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# Category Archives: Hydrodynamics

## The Gresho vortex

Notes on the Gresho vortex, chiefly the pressure field.

## Nuevo artículo, fecha: San Isidro 2017

Nuevo artículo aceptado: Extending linear finite elements to quadratic precision on arbitrary meshes, Daniel Duque, Pep Español, Jaime Arturo de la Torre (UNED, los dos últimos). Descargable gratuitamente hasta fin de febrero.

Fecha de "publicación" (en papel, supongo): 15 de mayo de 2017. ¡San Isidro!

## Taylor-Green vortex sheet, reduced units

The Taylor-Green vortex sheet is a solution to the 2D Navier-Stokes equations for an incompressible Newtonian fluid: $latex \frac{d \mathbf{u}}{d t}= \nu \nabla^2 \mathbf{u} – \nabla p/\rho$ where $latex \mathbf{u}$ is the velocity field, $latex p$ is the pressure,  $latex \nu=\mu/\rho$ is the kinematic viscosity, and $latex \rho$ is the fixed density of the fluid. The time derivative is a total derivative: $latex \frac{d \mathbf{u}}{d t} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$.

error:  $latex d \mathbf{u} = mathbf{u}$,

$latex \frac{d \mathbf{u}}{d t}$

It is common to choose parameters that simplify the equations, but that can obscure the role of the different parameters. In the following, I provide expressions with all relevant parameters included, with their physical dimensions. I later pass to dimensionless, or reduced, units, in terms of the Reynolds and Courant numbers.