The von Neumann stability analysis is a great tool to assess stabilty of discretizing schemes for PDEs. But too often, imho, the discussion gets blurred with trigonometric expressions. Here, I try to provide a shortcut.
BTW: it’s “fon no ee man”
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.
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 quick cheatsheet for OpenFOAM. In italics, things that are useful but not part of OpenFOAM proper. Interesting read: The OpenFOAM Technology Primer
Shortcuts to directories
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Here is what, for many, is a weak point in wikipedia: the sources. What is accepted as “true”. Here is a list of claims that may have been challenged in wikipedia, or not. See how many you get right!