Euler’s Equations

Euler derived the fundamental equations of an ideal fluid, that is, in the absence of friction (viscosity). They describe the conservation of momentum. We can derive from it the equation for the evolution of vorticity (Helmholtz equation). Euler’s equations have to be supplemented by the conservation of mass and by an equation of state (which relates density to pressure). Of special interest is the case of incompressible flow; when the fluid velocity is small compared to the speed of sound, the density may be treated as a constant. In this limit, Euler’s equations have scale invariance in addition to rotation and translation invariance. d’Alembert’s paradox points out the limitation of Euler’s equation: friction cannot be ignored near the boundary, nomatter how small the viscosity.

A physics-based turbocharger model for automotive diesel engine control applications

Control-oriented models of turbocharger processes such as the compressor mass flow rate, the compressor power, and the variable geometry turbine power are presented. In a departure from approaches that rely on ad hoc empirical relationships and/or supplier provided performance maps, models based on turbomachinery physics and known geometries are attempted. The compressor power model is developed using Euler’s equations of turbomachinery, where the gas velocity exiting the rotor is estimated from an empirically identified correlation for the ratio between the radial and tangential components of the gas velocity. The compressor mass flow rate is modeled based on mass conservation, by approximating the compressor as an adiabatic converging-diverging nozzle with compressible fluid driven by external work input from the compressor wheel. The variable geometry turbine power is developed with Euler’s equations, where the turbine exit swirl and the gas acceleration in the vaneless space are neglected. The gas flow direction into the turbine rotor is assumed to align with the orientation of the variable geometry turbine vane. The gas exit velocity is calculated, similar to the compressor, based on an empirical model for the ratio between the turbine rotor inlet and exit velocities. A power loss model is also proposed that allows proper accounting of power transfer between the turbine and compressor. Model validation against experimental data is presented.

Knotted solutions, from electromagnetism to fluid dynamics

Knotted solutions to electromagnetism and fluid dynamics are investigated, based on relations we find between the two subjects. We can write fluid dynamics in electromagnetism language, but only on an initial surface, or for linear perturbations, and we use this map to find knotted fluid solutions, as well as new electromagnetic solutions. We find that knotted solutions of Maxwell electromagnetism are also solutions of more general nonlinear theories, like Born–Infeld, and including ones which contain quantum corrections from couplings with other modes, like Euler–Heisenberg and string theory DBI. Null configurations in electromagnetism can be described as a null pressureless fluid, and from this map we can find null fluid knotted solutions. A type of nonrelativistic reduction of the relativistic fluid equations is described, which allows us to find also solutions of the (nonrelativistic) Euler’s equations.

Eroding dipoles and vorticity growth for Euler flows in : axisymmetric flow without swirl

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler’s equations in $\mathbb{R}^{3}$. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the $t^{4/3}$ growth mimics an exact solution of Euler’s equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analysed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures.