# Quick von Neumann stability analysis

von Neumann stability analysis is a great tool to assess the stabilty of discretizing schemes for PDEs. But too often, imho, the discussion is too convoluted. Here, I try to provide a shortcut.

In a nutshell:

• The standard approach involves Fourier techniques, involving (of course) complex numbers
• The real part of these numbers is analysed, with some trigonometric expression resulting, identifying the troublesome modes
• I claim this mode can be identified in advance, which makes the whole Fourier procedure unnecessary

BTW: it’s pronounced “fon no ee man”

# 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:

$\frac{d \mathbf{u}}{d t}= \nu \nabla^2 \mathbf{u} - \nabla p/\rho ,$

where $\mathbf{u}$ is the velocity field, $p$ is the pressure, $\nu=\mu/\rho$ is the kinematic viscosity, and $\rho$ is the fixed density of the fluid. The time derivative is a total derivative:

$\frac{d \mathbf{u}}{d t} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$

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.

# Sound waves with attenuation

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

$\frac{\partial }{\partial t} \mathbf{u} + \mathbf{u} \nabla \mathbf{u} = - \frac{1}{\rho} \nabla p + \frac{\mu}{\rho} \nabla^2 \mathbf{u} + \left(\frac{\lambda+\mu}{\rho}\right)\nabla (\nabla\cdot\mathbf{u}) ,$

where $\lambda$ is a Lamé viscosity coefficient. The bulk viscosity coeficient  is defined as $\zeta = \lambda + (2/3) \mu$. 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.

# OpenFOAM cheatlist

A quick cheatsheet for OpenFOAM. In italics, things that are useful but not part of OpenFOAM proper. Interesting read: The OpenFOAM Technology Primer

Sites

Shortcuts to directories

(type “alias” to reveal these)

• run (go to own’s running directory)
• foam
• foamfv
• foam3rdParty (hit <tab> for these longish commands!)
• tut
• app
• sol
• util
• lib
• src

Environment variables

• echo $FOAM_ <tab> (directories) • echo$WM_ <tab>  (building, aka compiling, settings)

Mesh

• Generation
• Import / export
• foamMeshToFluent
• fluentMeshToFoam, etc …
• Operations
• refineHexMesh
• transformPoints
• makeAxialMesh
• collapseEdges
• autoPatch
• mirorMesh
• Properties
• checkMesh

Fields

• setFields
• topoSet
• patchAverage
• patchIntegrate
• vorticity
• yPlusRAS
• yPlusLES
• boxTurb
• applyBoundaryLayer
• R
• wallShearStress

Help

• foamHelp (e.g. foamHelp boudary -field U)

Postprocessing

• sample
• paraview
• probeLocations

Solvers

• icoFoam
• interFoam
• many, many others

Running

• foamJob
• decomposePar
• reconstructPar
• mpirun
• nohup

Building

• foamNew source App …
• doxygen doxyfile
• gdb
• valgrind
• wmake
• wclean
• aliases for settings: wm32, wm64, wmSP, wmDP, wmSET, wmUNSET

# wikipedia: reliable sources

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!

• About the katakana Japanese syllabary:  in a manga, the speech of a foreign character or a robot may be represented by, for example, コンニチワ konnichiwa (“hello”) instead of the more usual hiragana こんにちは.
• About Hugo_Stiglitz (a character in Inglourious_Basterds): Fittingly, the character’s guitar riff theme is taken from Slaughter, a blaxploitation movie starring Jim Brown.
• In the article about the film Watchmen: Classical music excerpts include Wagner’s “Ride of the Valkyries” in a Vietnam battle (a reference to Apocalypse now) and Mozart’s Requiem at the resolution in Antarctica.
• Getting serious:  In the article about the well-known D’Alembert’s_paradox of hydrodynamics: Recently the following alternative resolution of d’Alembert’s paradox was claimed [7][8][9] … This is a different resolution from that by Prandtl and it is claimed to be supported by both theory, computation and experiment. Notice that Refs. 7, 8, and 9 correspond to two articles in peer-reviewed journals and a book published by Springer!