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Guide: Writing Equations in LaTeX on the CFD Online Forums

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Old   March 15, 2009, 15:58
Default Guide: Writing Equations in LaTeX on the CFD Online Forums
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This is a brief introduction about how to write equations in the CFD Online discussion forums. The equation functionality is based on LaTeX, a versatile mathematical typesetting system which is often used to write scientific papers and books. With LaTeX mathematical equations are written using normal text and this makes it very suitable for a text-based forum like this. The forum will automatically generate images of the LaTeX codes it finds and display them properly. LaTeX often gives very nice looking formulas. LaTeX can be a bit difficult to get started with, but those who know it tend to like it very much. Here is a simple example to illustrate how it works:
  • If you in your message write: [math]a = \sqrt{\gamma R T}[/math]

  • The forum will display: a = \sqrt{\gamma R T}
A usefull feature in the forum is that the LaTeX code used to write an equation can be seen when you hoover with the pointer over it. If you click on an equation a window will be opened displaying the LaTeX code used to write it. This makes it easy to learn from other people and to copy-and-paste equations from other posts.

You can see how all equations on this page have been written either by hoovering over them with the pointer or by clicking on them!

LaTeX can be used in two ways:
  • Normal equations written separately away from normal text: [math]LaTeX code[/math]

  • Inlined equations or symbols to be used inside a text: [imath]LaTeX code[/imath]
The difference is that inlined equations are scaled to fit better in normal text whereas normal equations are scaled to look good when written separately.

There is a special LaTeX reference tool avalable in the advanced editor which helps you to find the codes needed to write symbols, relations, operators etc. in LaTeX. To open the LaTeX reference click on the \sum symbol to the right just above the editor window.

Below follows a summary of the most commonly used symbols, relations, operators and functions as well as a few other frequently asked questions. To see how these symbols or equations were created just click on the one you are interested in and a window will be opened showing the code used to create it:


Symbols

Lowercase Greek letters: \alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta \theta \vartheta \iota \kappa \lambda \mu \nu \xi \pi \varpi \rho \varrho \sigma \varsigma \tau \upsilon \phi \varphi \chi \psi \omega

Capital Greek letters: \Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega

Operators and relations: \pm \mp \times \div \ast \star \bullet \circ \cdot \leq \ll \subset \geq \gg \equiv \sim \simeq \approx \neq \propto

Misc: \leftarrow \Leftarrow \rightarrow \Rightarrow \leftrightarrow \Leftrightarrow \forall \exists \partial \infty


Bracketing - () {} []

You can simply write the brackets you want. To get the correct size of the brackets you sometimes have to use the \left and \right keywords in front of the bracket.
For example, just writing parentheses gives (\frac{1}{2})^{n}, but with \left and \right keywords you would get \left(\frac{1}{2}\right)^{n}.


Exponents and Subscripts

a_i^n - [math]a_i^n[/math]

x_{i+1}^{2^{n+1}} - [math]x_{i+1}^{2^{n+1}}[/math]


Functions - Fractions, Radicals, Integrals, Sums, Limits, ...

\frac{1}{2} - [math]\frac{1}{2}[/math]

\sqrt{x+y} - [math]\sqrt{x+y}[/math]

\sqrt[n]{x} - [math]\sqrt[n]{x}[/math]

\int^\infty_0 f(x) \, dx - [math]\int^\infty_0 f(x) \, dx[/math]

\oint_\Gamma f(x) \, ds - [math]\oint_\Gamma f(x) \, ds[/math]

\sum_{i=1}^{\infty}\frac{1}{i} - [math]\sum_{i=1}^{\infty}\frac{1}{i}[/math]

\lim_{x\to\infty}\frac{1}{x} - [math]\lim_{x\to\infty}\frac{1}{x}[/math]


Trigonometric and Logarithmic Functions

Please use these instead of simply typing in "sin". That ensures that the function is typeset properly with the right font and spacing.

\cos^2 x + \sin^2 x = 1 - [math]\cos^2 x + \sin^2 x = 1[/math]

\tanh(c) - [math]\tanh(c)[/math]

\log n^2 - [math]\log n^2[/math]

\ln(e) - [math]\ln(e)[/math]


Spacing

Sometime you may want to add extra horizontal space between characters. The following codes can be used to add extra space:
  • a whitespace or no space at all gives a normal space (\alpha \beta)
  • \quad - a large space to separate equations or write conditions (i = 1, 2, 3) on the same line (\alpha \quad \beta)
  • \; - a thick space (\alpha \; \beta)
  • \: - a medium space (\alpha \: \beta)
  • \, - a thin space (\alpha \, \beta)
  • \! - a negative thin space (\alpha \! \beta)
Fluid Dynamics and CFD Related Examples
  • Navier-Stokes Equations using Tensor Notation:
\frac{\partial \rho}{\partial t} +
\frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0

\frac{\partial}{\partial t}\left( \rho u_i \right) +
\frac{\partial}{\partial x_j}
\left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0, \quad i=1,2,3

\frac{\partial}{\partial t}\left( \rho e_0 \right) +
\frac{\partial}{\partial x_j}
\left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0
  • \bf{k - \epsilon} Equations:
\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k

\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}
  • Reynolds Averaging:
\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt

\Phi' \equiv \Phi - \overline{\Phi}
  • Boussinesq Eddy Viscosity Assumption:
\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} k \delta_{ij} , or written more explicitly:

-\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{2}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} k \delta_{ij}
  • Rhie-Chow Interpolation:
a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{\nabla p}{V}

\sum\limits_{faces} \left[ {\frac{1}{a_p}\,H} \right]_{face} = \sum\limits_{faces} \left[ {\frac{1}{a_p }\,\frac{{\nabla p}}{V}} \right]_{face}

H = \sum\limits_{neighbours} {a_l } \vec v_l
  • Realisability and Schwarz' Inequality:
\left\langle{u'_\alpha u'_\alpha}\right\rangle \geq 0

\frac{\left\langle{u'_\alpha u'_\beta}\right\rangle}{ \left\langle{u'_\alpha u'_\beta}\right\rangle \left\langle{u'_\alpha u'_\beta}\right\rangle} \leq 1
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Old   October 30, 2009, 02:51
Default test
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c=sqrt(a^2 + b^2) testing
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Old   October 30, 2009, 03:03
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\lim_{x\to\infty}\frac{1}{x}

lim_{xtoinfty}frac{1}{x-1}
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Old   October 30, 2009, 03:04
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\int^\infty_0 f(x) \, dx
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Old   October 30, 2009, 03:06
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\int^\frac{\Pi}{2}_5 f(x) \, dx
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Old   December 31, 2010, 02:03
Default Useful website for generating LaTeX equations
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Enjoy: http://www.codecogs.com/latex/eqneditor.php

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Old   November 6, 2011, 23:56
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cool. i often use mathematica to write equations. then they can be transform to latex form.
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Old   April 14, 2013, 16:11
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test test

\frac{1}{2}
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Old   May 2, 2013, 22:15
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let me have a try:
-\frac{1}{2}\rho\overline{u'_i u'_j}+\overline{p' u'_j}=\frac{\mu_t}{\sigma_k}\frac{\partial k}{\partial x_j}

and this " -\frac{1}{2}\rho\overline{u'_i u'_j}+\overline{p' u'_j}=\frac{\mu_t}{\sigma_k}\frac{\partial k}{\partial x_j} "inline equation
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Old   June 4, 2013, 20:49
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test

a = \sqrt{\gamma R T}
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Old   October 24, 2013, 12:58
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test

\frac{\partial c} {\partial t} + \nabla . ({\vec u - u_s \frac{\vec g} {| g |}}) c=\nabla . ( {\nu_t \nabla c} )


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Old   May 6, 2014, 21:18
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why can't I use \^{\circ}C?
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Old   May 6, 2014, 21:22
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Quote:
Originally Posted by Betta View Post
why can't I use \^{\circ}C?
sorry. it works now.
^{\circ}C
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Old   December 9, 2014, 07:38
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k_Sgs= \left(\frac{\mu_Sgs}{\rho \, C_s \, \Delta}\right)^{2}

C_s= 0.1

Last edited by babakflame; December 9, 2014 at 08:44.
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Old   January 26, 2015, 10:21
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{\nabla \bullet \left( \rho c_p \vec u T \right)}={\nabla \bullet \left( \lambda_{e f f} \nabla T \right)}  + T {\nabla \bullet \left( \rho c_p \vec u \right)} - \frac{T}{\rho} \left( \frac{\partial \rho}{\partial t} \right)_p \vec u \bullet {\nabla p}

Last edited by Gary51075607; January 26, 2015 at 22:01.
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Old   January 29, 2015, 21:51
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Just to have a try
\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k
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Old   May 25, 2015, 13:49
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O + N_2 \rightleftharpoons  N + NO
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A good solution is one which does justice to the inner nature of the problem- Cornelius Lanczos in a letter to Albert Einstein on March 9, 1947

Last edited by flowAlways; May 25, 2015 at 17:42.
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Old   June 13, 2015, 10:18
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I cannot figure this out. Where is my syntax error?

\alpha\frac{\partial^2x}{\partial \xi^2}-2\beta \frac{\partial ^2x}{\partial \xi \partial \eta}+\gamma \frac{\partial ^2x}{\partial\eta^2}=0
\alpha\frac{\partial^2y}{\partial\xi^2}-2\beta\frac{\partial^2y}{\partial\xi\partial\eta}+\gamma\frac{\partial^2y}{\partial\eta^2}=0

Here's what I typed between the bracketed math and /math:

\alpha\frac{\partial^2x}{\partial \xi^2}-2\beta \frac{\partial ^2x}{\partial \xi \partial \eta}+\gamma \frac{\partial ^2x}{\partial\eta^2}=0

\alpha\frac{\partial^2y}{\partial\xi^2}-2\beta\frac{\partial^2y}{\partial\xi\partial\eta}+ \gamma\frac{\partial^2y}{\partial\eta^2}=0

Here the images of what displays on the codehogs.com online eqn editor:





I don't understand why I'm getting an error on the forum. It's frustrating.
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Old   June 15, 2015, 16:39
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Seems like the LaTeX functionality is broken after the server move on June 5. We will investigate and post a message in this thread once LaTeX is working again.
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Old   June 15, 2015, 16:58
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Quote:
Originally Posted by pete View Post
Seems like the LaTeX functionality is broken after the server move on June 5. We will investigate and post a message in this thread once LaTeX is working again.
Thanks! I ended up posting equations as images obtained from the online LaTeX editor on codecogs.com but I found out I can post a maximum of 10 images.
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