The Taylor-Green vortex sheet is a solution to the 2D Navier-Stokes equations for an incompressible Newtonian fluid:
where is the velocity field, is the pressure, is the kinematic viscosity, and is the fixed density of the fluid. The time derivative is a total derivative:
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.
For the time being, this is just a stub to upload stuff for a discussion in CFD-online!
The Rayleigh-Taylor instability is a well-known benchmarck for CFD codes. The idea is to start with two phases, on on top of the other, the lighter one being underneath. The interface is slightly perturbed, and this plume appears. I describe a quick and dirty way of getting this instability.
Just a simple derivation of the role of attenuation in the standard sound wave equation. Original work: Stokes, 1845.
Starting with the Navier-Stokes momentum equation
where is a Lamé viscosity coefficient. The bulk viscosity coeficient is defined as . The last term is often neglected, even in compressible flow, but sound attenuation is one of the few cases where it may have some influence. All viscosities are assumed to be constant, but in this case this is a safe assumption, since we are going to assume small departures about equilibrium values.
A very technical summary of the changes I had to introduce in CMakeLists.txt in order to include correctly the eigen libs, then the suitesparse libs.
Back to scientific computing. Lately, I have been using the Eigen libraries for linear algebra. They were recommended by the CGAL project, and they indeed share a common philosophy. Thanks to the rather high C++ level they can accomplish this sort of magic:
int n = 100;
VectorXd x(n), b(n);
// solve Ax = b
x = solver.solveWithGuess(b,x0);
Notice that A is a sparse matrix! I am next describing how to use this in order to solve the 1D Poisson equation.
Some notes about how to transfer tracks from bandhub, a site for musical collaborations, onto cubase 5 for processing, mixing, and all that. Also, some notes about audacity at the end. Continue reading