Boring stuff

Posted on 29 October, 2018 by ddcampayo

The previous equation is still too general, and a connection between stress and strain is still needed. Here we consider the case in which there is a linear relationship between both, which involves the coefficient of viscosity.

To begin with, let us consider a simple case in which a fluid is confined between two planes. One of them moves sideways with a certain speed $u_0$ , while the other is kept fixed. After a certain transient, some force is needed in order to keep this shearing. The simplest expression is

$F= \mu A \frac{u_0}{L}.$

The force is proportional to the area and to the velocity difference between the planes. It is also inversely proportional to their separation, L (this fact being the least obvious). Finally, a constant of proportionality is given by $\mu$ , the viscosity coefficient, or

simply “the viscosity”. This constant may vary with temperature, density, pressure, but the point with Newtonian fluids is that it does not vary with the velocity field (or its derivatives).

Later, in section …, this flow will be solved as a solution of the Navier-Stokes equations, the Couette flow. There, it will be shown that the velocity is everywhere in the direction of the force exerted on the upper plane, let us call it $x$, and varies linearly between the planes, in the y direction. Therefore, the only components of the strain rate tensor are $\epsilon_{xy} = \epsilon_{yx} = u_0 / ( 2 L )$ . We therefore have

$\tau_{xy} = \mu \epsilon_{xy}.$

With this in mind, let us look for a general relationship between $\tau$ and $\epsilon$ . This is much easier if we go to the principal strain axes. These are the coordinates on which the strain rate is diagonal. Such coordinate system always exist, since the strain rate tensor is symmetric. Notice that in these system strains are not due to shear, only to dilations.

Better with stock pictures

Posted on 29 October, 2018 by ddcampayo

“A connection is needed”. Photo by Pixabay on Pexels.com

Holy cow, is honey viscous or what. Photo by Pixabay on Pexels.com

$F= \mu A \frac{u_0}{L}.$

action balls black and white illustration

Newton’s craddle. Another of this guy’s creations. Photo by Pixabay on Pexels.com

simply “the viscosity”. This constant may vary with temperature, density, pressure, but the point with Newtonian fluids is that it does not vary with the velocity field (or its derivatives).

What a wave. Its strain tensors must be on fire. Photo by Emiliano Arano on Pexels.com

$\tau_{xy} = \mu \epsilon_{xy}.$

With this in mind, let us look for a general relationship between $\tau$ and $latex

What a bunch of math. This is so hard. Photo by Lum3n.com on Pexels.com

\epsilon$. This is much easier if we go to the principal strain axes. These are the coordinates on which the strain rate is diagonal. Such coordinate system always exist, since the strain rate tensor is symmetric. Notice that in these system strains are not due to shear, only to dilations.

Tired of that “TeX” look?

Posted on 13 March, 2018 by ddcampayo

TeX has been using computer modern (CM) font since its inception. But that “TeX” look may become a bit tiring. Of course, TeX is a typesetting engine, it is not limited to CM fonts. On the other hand, there aren’t so many fonts around for both the text and the math. (If you have no math, xeTex makes it easy to use most fonts you can imagine, including the Microsoft and google families).

I found a very clear review of existing alternatives at the Font usage post, by Ryosuke Iritani (入谷亮介). I have taken his suggestions and created a gallery, with a simple sample of text and equations.

More elegant Palatino

\usepackage[sc]{mathpazo}
\linespread{1.05} % Palladio needs more leading (space between lines)
\usepackage[T1]{fontenc}

Kpfonts (Palatino-like)

\usepackage{kpfonts}

mathpazo

Libertine

Used e.g. in Wikipedia on each sectioning

\usepackage{libertine}
\usepackage{libertinust1math}
\usepackage[T1]{fontenc}

STIX

Scientific and Technical Information Exchange; Times-based but much more elegant than txfonts package.

\usepackage[T1]{fontenc}
\usepackage{stix}

styx

Garamond

It’s a bit thin and less friendly

\usepackage[urw-garamond]{mathdesign}
\usepackage[T1]{fontenc}

Utopia (Adobe)

\usepackage[adobe-utopia]{mathdesign}
\usepackage[T1]{fontenc}

Charter

\usepackage[charter]{mathdesign}

Crimson (with math support)

\usepackage[T1]{fontenc}
\usepackage{cochineal}
\usepackage[cochineal,varg]{newtxmath}

Baskervald

Baskerville-based, thicker font

\usepackage[lf]{Baskervaldx} % lining figures
\usepackage[bigdelims,vvarbb]{newtxmath} % math italic letters from Nimbus Roman
\usepackage[cal=boondoxo]{mathalfa} % mathcal from STIX, unslanted a bit
\renewcommand*\oldstylenums[1]{\textosf{#1}}

Helvetica

So far, the only font not included the Iritani’s Font usage post!

\usepackage{helvet}
\usepackage{sansmath}

\usepackage{titlesec}  % this enforces helvetica in section and chapter titles
\titleformat{\chapter}[display]
  {\normalfont\sffamily\huge\bfseries}
  {\chaptertitlename\ \thechapter}{20pt}{\Huge}
\titleformat{\section}
  {\normalfont\sffamily\Large\bfseries}
  {\thesection}{1em}{}

% In main text, at the beginning:
\fontfamily{phv}\selectfont

% before the first equation:
\sansmath

helvetica

The code

All the above was produced with variations of this file. I just run latex on it, then dvips to get a ps file, which I then crop and export as PNG using the GIMP. Of course, depending on the system, some LaTeX packages may be needed, as well as fonts (I had to install urw-garamond on my arch linux system, for example.)

\documentclass{article}

\newcommand{\bfr}{\mathbf{r}}
\newcommand{\bfu}{\mathbf{u}}
\newcommand{\bfq}{\mathbf{q}}

\usepackage{amsmath}

\usepackage{libertine}
\usepackage{libertinust1math}
\usepackage[T1]{fontenc}

\usepackage{lipsum}% for filler text

\begin{document}

\section{A section}

\lipsum[10]

Equations:

\begin{equation}
\frac{d \mathbf{u}}{d t} = - \nabla p + \nu \nabla^2 \mathbf{u},
\end{equation}

\begin{equation}
\begin{split}
E &= m c^{2},\\
T &= 2\pi \sqrt{\frac{m}{k}}
\end{split}
\end{equation}

\begin{equation}
\iint \phi = - \oint p
\end{equation}
\end{document}

Plotting 2D column-shaped results with python

Posted on 6 March, 2018 by ddcampayo

Ok, so you have your computational output of a set of 2D points. You have been lazy and done the obvious stuff: arrange them in columns, with the first one being the x coordinates, second one the y coordinate, then come the fields, which may be scalar (one column each), or vector (two columns each, one for each coordinate). How to visualize them?

With python, start with

ipython --pylab

Then, read the data

In [4]: dt = loadtxt( ‘1/mesh.dat’ )

In [5]: shape( dt )
Out[5]: (1024, 27)

Notice the last command tells us we have 1024 data points, and 27 fields (well, 25 + positions). For convinience, assign columns to arrays:

In [6]: x=dt[:,0]; y=dt[:,1]; al=dt[:,4]

Now x and y are positions, and “al” is the scalar field for the fifth column (number 4, since counters start at 0 in python).

To visualize the positions,

In [7]: scatter( x , y )

A scalar field may be visualized with a color map:

In [9]: scatter( x , y , c = al )

The “c=” means the color is taken from field al. One may fiddle with colormaps and symbol sizes:

In [9]: scatter( x , y , c = al , cmap= plt.cm.Blues, s=8 )

To know the range we are plotting, produce a color bar:

In [19]: colorbar()

Remember each plotting is overlaid on the previous one, so it is necessary to blank the plot from time to time:

In [11]: clf()

For vector fields, assign coordinates to two separate arrays:

In [20]: vx=dt[:,8]; vy=dt[:,9]

Then, use “quiver” to get a vector plot:

In [22]: quiver( x, y, vx , vy )

A markdown test

Posted on 26 February, 2018 by ddcampayo

This is a test of markdown blog writing. The writing comes straight from my website on CFD methods. These were written in markdown under reveal.js, for quick and nice lecture slides. Some changes had to be made:

LaTeX must start as “dollar sign latex” … “dollar”
Links to local files (such as pictures) don’t work
Lists (such as this one) do not seem to work well

Muy a menudo, se parte de las EDPs, conocidas, por ejempo:

$\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$

Estas se discretizan: sustituyendo las derivadas por diferencias.

Sin embargo, este es un proceso de ida y vuelta, porque
las EDPs se deducen a nivel discreto.

Deducción

Se suponen cambios de un campo $u$ sólo
en la dirección $x$

Esquema de convección en 1D

conveccion_1d

parts_bw_values_250

El cambio en la cantidad total $A \Delta x \, u_i$ será:

$\frac{d }{d t} (A \Delta x \, u_i ) = \Phi_{i-1/2} - \Phi_{i+1/2}$

Flujos, convección

Antes los flujos por las caras venían dados por:

$\Phi_{i-1/2} = A c \, u_{i-1/2}$

$\Phi_{i+1/2} = A c \, u_{i+1/2}$

$\frac{d }{d t} (A \Delta x \, u_i ) = A c \, u_{i-1/2} - A c \, u_{i+1/2}$

Sound waves with attenuation

Posted on 20 November, 2015 by ddcampayo

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.

Continue reading →

CGAL, eigen, suitesparse, cmake tweaking

Posted on 11 November, 2015 by ddcampayo

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.

Continue reading →

Sparse Poisson problem in eigen

Posted on 25 March, 2015 by ddcampayo

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);
  SpMat A(n,n);

  fill_A(A);

  fill_b(b);

  // solve Ax = b
  Eigen::BiCGSTAB<SpMat> solver;
  //ConjugateGradient<SpMat> solver;

  solver.compute(A);

  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.

Continue reading →

OpenFOAM cheatlist

Posted on 26 February, 2015 by ddcampayo

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
- blockMesh
- snappyHexMesh
- funkySetFields (part of the swak4Foam project, together with groovyBC)
- foamyHexMesh
- foamyQuadMesh
- cfMesh (provided by creative fields )
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

Sorting eivenvectors in octave

Posted on 20 February, 2015 by ddcampayo

A brief note. Doing ee= eig(m); produces a vector with all eigenvalues of matrix m. These are unsorted, so ee=sort(eig(m)); produces a vector with a sorted list of all eigenvalues of matrix m.

If we want the eigenvectors, [vv,eed]=eig(m); produces a diagonal matrix eed whose elements are the same as ee, and are unsorted. Who to produce a sorted vector with the eigenvalues, and re-order the eigenvectors accordingly? Thus:

[vv,eed]=eig(m);

[ee,perm] = sort(diag(eed));

vv=vv(:,perm);

Daniel Duque Campayo

web-ish blog

Category Archives: computing

Boring stuff

Better with stock pictures

Tired of that “TeX” look?

More elegant Palatino

Kpfonts (Palatino-like)

Libertine

STIX

Garamond

Utopia (Adobe)

Charter

Crimson (with math support)

Baskervald

Helvetica

The code

Plotting 2D column-shaped results with python

A markdown test

Deducción

Flujos, convección

Sound waves with attenuation

CGAL, eigen, suitesparse, cmake tweaking

Sparse Poisson problem in eigen

OpenFOAM cheatlist

Sorting eivenvectors in octave