9 research outputs found
Iterated upwind schemes for gas dynamics
A class of high-resolution schemes established in integration of anelastic equations
is extended to fully compressible flows, and documented for unsteady (and
steady) problems through a span of Mach numbers from zero to supersonic. The
schemes stem from iterated upwind technology of the multidimensional positive
definite advection transport algorithm (MPDATA). The derived algorithms employ
standard and modified forms of the equations of gas dynamics for conservation of
mass, momentum and either total or internal energy as well as potential temperature.
Numerical examples from elementary wave-propagation, through computational
aerodynamics benchmarks, to atmospheric small- and large-amplitude acoustics
with intricate wave-flow interactions verify the approach for both structured
and unstructured meshes, and demonstrate its flexibility and robustness
Asymptotics, structure, and integration of sound-proof atmospheric flow equations
Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran's pseudo-incompressible model, are shown here to be structurally very
close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing @t in the mass continuity equation and slightly modifying the gravity term, whereas the pseudoincompressible model results from dropping @tp from the pressure equation.
For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density
flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible
flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere-ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (2007).
Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura & Phillips model, which assumes weak stratication of the order of the Mach number squared, can be obtained as a
leading-order model from systematic low Mach number asymptotic analysis.
Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the
descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work
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An Anelastic Allspeed Projection Method for Gravitationally Stratified Flows
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An Anelastic Allspeed Projection Method for Gravitationally Stratified Flows
This paper looks at gravitationally-stratified atmospheric flows at low Mach and Froude numbers and proposes a new algorithm to solve the compressible Euler equations, in which the asymptotic limits are recovered numerically and the boundary conditions for block-structured local refinement methods are well-posed. The model is non-hydrostatic and the numerical algorithm uses a splitting to separate the fast acoustic dynamics from the slower anelastic dynamics. The acoustic waves are treated implicitly while the anelastic dynamics is treated semi-implicitly and an embedded-boundary method is used to represent mountain ranges. We present an example that verifies our asymptotic analysis and a set of results that compares very well with the classical gravity wave results presented by Durran