33 research outputs found
A General Method to Compute Numerical Dispersion Error
This article presents a new methodology to compute numerical dispersion error. The analysis here presented is not restricted to uniform structured meshes nor linear discrete operators as it does not rely on sinusoids to compute the associated error. When using uniform meshes, the results obtained with the present method collapse onto the obtained with the classic one via an easy change of basis. If non-uniform meshes are used, a new kind of results are obtained which shed some light onto the role stretching has on dispersion error.This work has been financially supported by the Ministerio de EconomĂa y Competitividad, Spain (No. ENE2017-88697-R). J.R.P. is supported by a FI-DGR 2015 predoctoral contract financed by Generalitat de Catalunya, Spain.Peer ReviewedPostprint (published version
A general framework for the evaluation of shock-capturing schemes
We introduce a standardized procedure for benchmarking shock-capturing schemes which is intended to go beyond traditional case-by-case analysis, by setting objective metrics for cross-comparison of flow solvers. The main idea is that use of shock-capturing schemes yields both distributed errors associated with propagation of wave-like disturbances in smooth flow regions, and localized errors at shocks where nonlinear numerical mechanisms are most active. Our standardized error evaluation framework relies on previous methods of analysis for the propagation error with associated cost/error mapping, and on novel analysis of the shock-capturing error based on a model scalar problem. Amplitude and phase errors are identified for a number of classical shock-capturing schemes with different order of accuracy. Whereas all schemes are found to be, as expected, first-order accurate at shocks, quantitative differences are found to be significant, and we find that certain schemes in wide use (e.g. high-order WENO schemes) may yield substantial over-amplification of incoming disturbances at shocks. Illustrative calculations are also shown for the 1D Euler equations, which support sufficient generality of the analysis, although nonlinearity suggests caution in straightforward extrapolation to other flow cases
Eulerian-Lagrangian method for simulation of cloud cavitation
We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation
in a compressible liquid. The method is designed to capture the strong,
volumetric oscillations of each bubble and the bubble-scattered acoustics. The
dynamics of the bubbly mixture is formulated using volume-averaged equations of
motion. The continuous phase is discretized on an Eulerian grid and integrated
using a high-order, finite-volume weighted essentially non-oscillatory (WENO)
scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles
at the sub-grid scale, each of whose radial evolution is tracked by solving the
Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid
as the void fraction by using a regularization (smearing) kernel. In the most
general case, where the bubble distribution is arbitrary, three-dimensional
Cartesian grids are used for spatial discretization. In order to reduce the
computational cost for problems possessing translational or rotational
homogeneities, we spatially average the governing equations along the direction
of symmetry and discretize the continuous phase on two-dimensional or
axi-symmetric grids, respectively. We specify a regularization kernel that maps
the three-dimensional distribution of bubbles onto the field of an averaged
two-dimensional or axi-symmetric void fraction. A closure is developed to model
the pressure fluctuations at the sub-grid scale as synthetic noise. For the
examples considered here, modeling the sub-grid pressure fluctuations as white
noise agrees a priori with computed distributions from three-dimensional
simulations, and suffices, a posteriori, to accurately reproduce the statistics
of the bubble dynamics. The numerical method and its verification are described
by considering test cases of the dynamics of a single bubble and cloud
cavitaiton induced by ultrasound fields.Comment: 28 pages, 16 figure