Guillermo Hauke

Stabilized finite element methods

We give a brief overview of stabilized finite element methods and illus- trate the developments applied to the advection-diffusion equation.

A comparative study of different sets of variables for solving compressible and incompressible flows

A globally conservative Galerkin/least-squares formulation which attains correct shock structure is developed for any choice of variables. Only the choice of entropy variables satisfies exactly the discrete Clausius-Duhem inequality without any dissipative mechanisms, whereas for the rest of the variables, artificial diffusion is required to guarantee entropy production. The limit of the formulation is well defined for entropy variables and the primitive variables (p, u, T), leading to conservative incompressible formulations. The approach is stable for any continuous interpolations, both for compressible and incompressible flows. A comparative study of different variables is performed, indicating that entropy variables and the primitive variables (p, u, T) possess the most attributes for practical problem solving.

A simple subgrid scale stabilized method for the advection-diffusion-reaction equation

Abstract A subgrid scale method has been developed for the advection–diffusion-reaction equation with a simple intrinsic time-scale parameter. The proposed formulation combines very good accuracy and stability in all the physical regimes of the equation, that is, in the exponential regime for both, negative and positive source terms, and in the propagation regime. Numerical examples confirm the robustness and accuracy of the method.

Longitudinal instabilities in an air-blasted liquid sheet

An experimental and numerical study has been performed to improve the understanding of the air/liquid interaction in an air-blasted breaking water sheet. This research is focused in the near eld close to the exit slit, because it is in this region where instabilities develop and grow, leading to the sheet breakup. In the experiments, several relevant parameters were measured including the sheet oscillation frequency and wavelength, as well as the droplet size distribution and the amplication growth rate. The flow was also investigated using linear instability theory. In the context of existing papers on instability analysis, the numerical part of this work presents two unique features. First, the air boundary layer is taken into account, and the eects of air and liquid viscosity are revealed. Second, the equations are solved for the same parameter values as those in the experiments, enabling a direct comparison between calculations and measurements; although qualitatively the behaviour of the measured variables is properly described, quantitative agreement is not satisfactory. Limitations of the instability analysis in describing this problem are discussed. From all the collected data, it is conrmed that the oscillation frequency strongly depends on the air speed due to the near-nozzle air/water interaction. The wave propagates with accelerating interface velocity which in our study ranges between the velocity of the water and twice that value, depending on the air velocity. For a xed water velocity, the oscillation frequency varies linearly with the air velocity. This behaviour can only be explained if the air boundary layer is considered.

Variational subgrid scale formulations for the advection–diffusion-reaction equation

Abstract The exact variational multiscale (VMS) and the subgrid scale (SGS) methods have been developed for the advection-reaction and the advection–diffusion-reaction equations. From the element Greens function, approximate intrinsic time scale parameters have been derived for these cases and are shown to be similar to other expressions obtained in the literature out of the maximum principle and convergence/error analysis. The methods have been compared with typical stabilized finite element methods. As expected, the VMS is nodally exact for the one-dimensional case.

Simple stabilizing matrices for the computation of compressible flows in primitive variables

The key ingredient that balances stability and accuracy in stabilized formulations is the parameter of intrinsic time-scales. For multi-dimensional hyperbolic systems of equations, this parameter is a matrix and the available expressions for its computation involve the solution of an eigenvalue problem, which can be tedious or cpu time consuming. Thus, for formulations based on primitive variables including pressure, a couple of simple stabilizing matrices are presented which are easy to implement and cpu-economic. Numerical evaluations show the performance of the various choices.

A symmetric formulation for computing transient shallow water flows

Abstract The shallow water equations fall into the category of symmetric advective-diffusive systems with source terms. These types of equations can be very effectively solved using stabilized methods, such as SUPG and GLS. A semi-discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows. Special care has been taken in the design of the operators in order to improve the performance for unsteady calculations. The solution is advanced in time via a predictor multi-corrector algorithm which includes as special cases the second-order trapezoidal rule, a first-order ‘explicit’ method and a first-order implicit method, equivalent to the constant-in-time element of the space-time formulation.

COMBINING ADJOINT STABILIZED METHODS FOR THE ADVECTION-DIFFUSION-REACTION PROBLEM

Computational methods for the advection-diffusion-reaction transport equation are still a challenge. Although there exist globally stable methods, oscillations around sharp layers such as boundary, inner and outflow layers, are typical in multi-dimensional flows. In this paper a variational formulation that combines two types of stabilization integrals is proposed, namely an adjoint stabilization and a gradient adjoint stabilization. Two free parameters are chosen by imposing one-dimensional superconvergence. Then, when applied to multi-dimensional flows, the method presents better local stability than the present stabilized methods. Furthermore, in the advective-diffusive limit and for piecewise linear functional spaces, the method recovers the classical SUPG method.

Liquid compressibility effects during the collapse of a single cavitating bubble.

The effect of liquid compressibility on the dynamics of a single, spherical cavitating bubble is studied. While it is known that compressibility damps the amplitude of bubble rebounds, the extent to which this effect is accurately captured by weakly compressible versions of the Rayleigh-Plesset equation is unclear. To clarify this issue, partial differential equations governing conservation of mass, momentum, and energy are numerically solved both inside the bubble and in the surrounding compressible liquid. Radiated pressure waves originating at the unsteady bubble interface are directly captured. Results obtained with Rayleigh-Plesset type equations accounting for compressibility effects, proposed by Keller and Miksis [J. Acoust. Soc. Am. 68, 628-633 (1980)], Gilmore, and Tomita and Shima [Bull. JSME 20, 1453-1460 (1977)], are compared with those resulting from the full model. For strong collapses, the solution of the latter reveals that an important part of the energy concentrated during the collapse is used to generate an outgoing pressure wave. For the examples considered in this research, peak pressures are larger than those predicted by Rayleigh-Plesset type equations, whereas the amplitudes of the rebounds are smaller.

Influence of the accommodation coefficient on nonlinear bubble oscillations.

This paper numerically investigates the effect of mass transfer processes on spherical single bubble dynamics using the Hertz-Langmuir-Knudsen approximation for the mass flux across the interface. Bubble behavior, with and without mass transfer, is studied for different values of pressure wave amplitude and frequency, as well as initial bubble radius. Whereas mass transfer processes do not seem to play a significant role on the bubble response for pressure amplitudes smaller than 0.9 atm, they appear to have an important effect when the amplitude is greater than or equal to 1 atm. For the later case, where the minimum liquid pressure reaches values around its vapor pressure, the importance of mass transfer depends on frequency. For frequencies in the 10(3)-10(5) Hz range and initial bubble radii of the order of tens of microns, bubble implosions with and with no mass transfer are significantly different; smaller radii display a lower sensitivity. In this regime, accurate model predictions must, therefore, carefully select the correct value of the accommodation coefficient. For frequencies greater than 10(5) Hz, as a first approximation mass transfer can be ignored.