Latex Equations Cribsheet

Introduction

This page provides some introductory material for writing LaTex equations for the QNET Wiki. It is not intended to be a thorough introduction to LaTex, but is to provide some guidelines, share good practice advice and other information that aids the authoring of LaTex in this Wiki.

Editing Latex

To add an equation to an article click on the "Mathematical formula" button in the Edit toolbar.

Alternatively, use the math element in the Edit page of the article as follows:

<math>V = frac{4}{3} \pi R^{3}</math>

This should generate the expression:

$V={\frac {4}{3}}\pi R^{3}$

CFD Equations and Expressions

There are many commonly used CFD equations and expressions used in the QNET Wiki and some of the commonly used ones are listed here. These can be either copied verbatim or used as templates for similar equations, etc.

Basic Expressions

Basic expressions here.

Reynolds Number

${\mbox{Re}}={\frac {\rho {\overline {u}}d}{\mu }}$

Prandl Number

${\mbox{Pr}}={\frac {C_{p}\mu }{\lambda }}$

Basic Flow Equations

Mass Continuity Equation

${\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x_{j}}}(\rho u_{j})=0$

Eulers Equation

$\rho {\frac {D\mathbf {u} }{Dt}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} .\nabla \mathbf {u} \right)=-\nabla P$

Navier Stokes Equations

${\frac {\partial }{\partial t}}(\rho u_{i})+{\frac {\partial }{\partial x_{j}}}(\rho u_{i}u_{j}+p\delta _{ij}-\tau _{ji})=0$

where in the case of a Newtonian fluid:

$\tau _{ij}=2\mu S_{ij}^{*}$

and

$S_{ij}^{*}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)-{\frac {1}{3}}{\frac {\partial u_{k}}{\partial x_{k}}}\delta _{ij}$

Energy Transport Equation

${\frac {\partial }{\partial t}}(\rho e_{0})+{\frac {\partial }{\partial x_{j}}}(\rho u_{j}e_{0}+u_{j}p+q_{j}-u_{i}\tau _{ij})=0$

The heat flux $q_{j}$ is given by:

$q_{j}=-\lambda {\frac {\partial T}{\partial x_{j}}}\equiv -C_{p}{\frac {\mu }{\mbox{Pr}}}{\frac {\partial T}{\partial x_{j}}}$

Equations of State

Ideal Equation of State

$p=\rho RT$

Turbulence Equations

Reynolds Stress Transport Equation

${\frac {\partial {\overline {u_{i}}}}{\partial t}}+{\overline {u_{j}}}{\frac {\partial {\overline {u_{i}}}}{\partial x_{j}}}=-{\frac {1}{\rho }}{\frac {\partial {\overline {p}}}{\partial x_{i}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial x_{j}}}\left(\mu {\frac {\partial {\overline {u_{i}}}}{\partial x_{j}}}-\rho {\overline {u_{i}\prime u_{j}\prime }}\right)$

Standard Two Equation Model

Kinematic eddy viscosity:

$\nu _{T}=C_{\mu }k^{2}/\epsilon$

Turbulence Kinetic Energy transport equation:

${\frac {\partial k}{\partial t}}+U_{j}{\frac {\partial k}{\partial x_{j}}}=\tau _{ij}{\frac {\partial U_{i}}{\partial x_{j}}}-\epsilon +{\frac {\partial }{\partial x_{j}}}\left[(\nu +\nu _{T}/\sigma _{k}){\frac {\partial k}{\partial x_{j}}}\right]$

Dissipation Rate transport equation:

${\frac {\partial \epsilon }{\partial t}}+U_{j}{\frac {\partial \epsilon }{\partial x_{j}}}=C_{\epsilon 1}{\frac {\epsilon }{k}}\tau _{ij}{\frac {\partial U_{i}}{\partial x_{j}}}-C_{\epsilon 2}{\frac {\epsilon ^{2}}{k}}+{\frac {\partial }{\partial x_{j}}}\left[(\nu +\nu _{T}/\sigma _{\epsilon }){\frac {\partial \epsilon }{\partial x_{j}}}\right]$

Coefficients and Auxilliary Relations:

$C_{\epsilon 1}=1.44$||$C_{\epsilon 2}=1.92$||$C_{\mu }=0.09$ $\sigma _{k}=1.0$

$\sigma _{\epsilon }=1.3$

where

$\omega =\epsilon /{C_{\mu }k}$

and

$l=C_{\mu }k^{3/2}/\epsilon$