Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier-Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

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LASER induced breakdown spectroscopy (LIBS) has been used for the quantitative analysis of Cu–Ni alloy of known composition (75% Cu, 25% Ni) using the one line calibration free-LIBS (OLCF-LIBS), self-calibration-LIBS (SC-LIBS), calibration free LIBS (CF-LIBS), time of flight-mass spectroscopy (TOF-MS), energy dispersive X-ray spectroscopy (EDX) and X-ray fluorescence spectroscopy (XRF). For the LIBS-based studies, the plasma was generated by focusing the beam of a Q-switched Nd:YAG laser (532 nm, pulse energy about 200 mJ, 5 ns pulse duration) while the sample was placed in air at an atmospheric pressure. Plasma temperature about (9500 ± 300) K was calculated by the Boltzmann plot method using the neutral lines of Cu and Ni whereas the electron number density was calculated (2.0 ± 0.5) × 1016 cm−3 from the Stark broadening of an isolated Cu line as well as using the relative intensities of the neutral and singly ionized optically thin lines in the Saha–Boltzmann equation. The elemental compositions determined by different LIBS methods and standard techniques are; OLCF-LIBS (69% Cu and 31% Ni), SC-LIBS (72% Cu and 28% Ni), CF-LIBS (74% Cu and 26% Ni), TOF (74% Cu and 26% Ni), EDX (75% Cu and 24.5% Ni), XRF (73% Cu and 24.7% Ni), and LA-TOF (74% Cu and 26% Ni). It is demonstrated that the CF-LIBS method gives compositions comparable with that determined by LA-TOF, EDX, or XRF, which is also in agreement with the certified reported composition.