A nonlinear, compressible, spectral collocation code is employed to examine gravity wave breaking in two and three spatial dimensions. Two‐dimensional results exhibit a structure consistent with previous efforts and suggest wave instability occurs via convective rolls aligned normal to the gravity wave motion (uniform in the spanwise direction). Three‐dimensional results demonstrate, in contrast, that the preferred mode of instability is a series of counterrotating vortices oriented along the gravity wave motion, elongated in the streamwise direction, and confined to the region of convective instability within the wave field. Comparison of the two‐dimensional results (averaged spanwise) for both two‐ and three‐dimensional simulations reveals that vortex generation contributes to much more rapid wave field evolution and decay, with rapid restoration of near‐adiabatic lapse rates and stronger constraints on wave energy and momentum fluxes. These results also demonstrate that two‐dimensional models are unable to describe properly the physics or the consequences of the wave breaking process, at least for the flow parameters examined in this study. The evolution and structure of the three‐dimensional instability, its influences on the gravity wave field, and the subsequent transition to quasi‐isotropic small‐scale motions are the subjects of companion papers by Fritts et al. (this issue) and Isler et al. (this issue).

Companion papers by Andreassen et al. (this issue) and Fritts et al. (this issue) introduced a nonlinear, compressible, spectral collocation code and applied it to studies of gravity wave breaking in two and three dimensions. The former showed the two simulations to differ dramatically in the mode of instability and in its implications for the wave and mean flow evolutions. The latter considered in detail the structure and energetics of the instability and its influences via eddy transports of momentum and heat. This paper addresses the instability structure and evolution at late times, focusing specifically on secondary instability, vortex breakdown, and the transition to isotropic structure. These results exhibit several distinct behaviors, depending on the local environment. In the presence of weak environmental shears, vortex breakdown occurs through mutual interactions which cause a gradual nonlinear evolution toward smaller scales of motion. Where wave and mean shears are strong, vortex breakdown is accelerated by dynamical instability processes at small scales which modulate strongly the vortex structures due to wave instability. Spectral results suggest that our simulation has described the transition from two‐dimensional laminar wave motions to three‐dimensional isotropic small‐scale structure.

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One of the more promising recent approaches to turbulence modelling is the Variational Multiscale Large Eddy Simulation (VMS LES) method proposed by Hughes et al. [Comp. Visual. Sci., vol. 3, pp. 47-59, 2000]. This method avoids several conceptual issues of traditional filter-based LES by employing a priori scale partitioning in the discretization of the Navier-Stokes equations.Most applications of VMS LES reported to date have been based on hierarchical bases, in particular global spectral methods, in which scale partitioning is straightforward. In the present work we describe the implementation of the methodology in a three-dimensional high-order spectral element method with a nodal basis. We report results from coarse grid simulations of turbulent channel flow at different Reynolds numbers to assess the performance of the model.

SUMMARYThis paper reports the results of spectral element simulations of natural convection in two-dimensional cavities. In particular, a detailed comparison is performed with the reference data for the 8:1 cavity at Ra = 3:4 × 10 5 recently described by Christon et al. [Int. J. Numer. Methods Fluids 2002; 40:953 -980]. The Navier-Stokes equations augmented by the Boussinesq approximation to represent buoyancy e ects are solved by a numerical method based on a spectral element discretization and operator splitting. The computed solutions agree closely with the reference data for both the square and the rectangular cavity conÿgurations.