Solution of the Navier‐Stokes equations by the spectral‐difference method

Abstract

The Jacobi polynomial collocation method is extended to two and three dimensions by superimposing the individual one-dimensional expansions. Two innovative ideas are introduced in this article.The first is the cornerless/edgeless computational grid, and the second is the modified compressibility method, which is an iterative pressure solver. To evaluate the new method's applicability in solving the Navier-Stokes equation, the lid-driven cavity flow problem was solved. Two configurations were considered, the square cavity and the rectangular cavity with an aspect ratio of 2. A comparison of the center-line velocity profiles from two-and three-dimensional simulations at a Reynolds number of 1000 is provided for each of the configurations. The center-line velocity comparisons showed good quantitative agreement with previous studies.

A Pseudospectral Σ -Transformation Model of 2-D Nonlinear Waves

“…In the last decade, a pseudospectral matrix-element method (PSME) was established by Ku & Hatziavramidis (1985) and Ku et al (1987aKu et al ( , b, 1989. A similar idea was provided by Cortes & Miller (1994) as the spectral-di!erence method. The latter requires that separate one-dimensional expansions be performed for each relevant axis, and the "nal form of the expansion is the result of the superposition of the one-dimensional results.…”

Pseudospectral element model for free surface viscous flows

“…Consider a 2D cavity filled with fluid and covered by a horizontal moving lid. Rectangular cavity flow has been studied by many researchers (Burggraf, 1966; Nallasamy and Krishna Prasad, 1977; Ghia et al , 1982; Ku et al , 1987; Cortes and Miller, 1994). According to their results, the cavity flow has a main vortex in the centre and two secondary vortices at the lower corners respectively for cases with Reynolds number less than 1,000.…”

“…After Chorin (1968) proposed the artificial compressibility method, a series of compressibility methods were used for solving the time‐dependent Navier‐Stokes equations and the continuity equation (Puckett, 1997). One example is the modified compressibility method established by Cortes and Miller (1994). The main idea behind the modified compressibility method is to improve the phase accuracy of time‐dependent problems.…”

“…The crucial feature of this method is that explicit discretisation formulas for higher‐derivatives are obtained based on the assumption that the function behaviour is accurately represented by a Chebyshev series spanning the global domain. Moreover, Cortes and Miller (1994) extended the CPSM to the spectral‐difference method which can use other polynomials like Legendre polynomials as trial functions. The foregoing researchers applied their pseudospectral approaches to cavity driven flow, Couette‐Taylor flow and uniform flow past a circular cylinder, and obtained accurate results in comparison with other schemes.…”

A three-dimensional PSME model for rotating flows in an annular cavity

“…The present study makes use of a two-step modified compressibility method (Cortes and Miller, 1994) to calculate the pressure. In the first step, the continuity equation is modified to…”