# Total derivative manipulation

• May 11th 2009, 09:37 AM
james
Total derivative manipulation
Hey guys, I think I'm making this harder than it needs to be but anyway,

I have two transport equations, one in k (the turbulent kinetic energy) and one in $\displaystyle \phi$ (a dimensionaly independant variable). So two equations in the form
$\displaystyle \frac{Dk}{Dt} = ...$
$\displaystyle \frac{D \phi}{Dt} = ...$

I am also given a relation to $\displaystyle \epsilon$ (the rate of dissipation of turbulent kinetic energy) which is
$\displaystyle \phi = k^m \epsilon^n$
where m and n are constants.

I need to get the equation in $\displaystyle \phi$ into one in $\displaystyle \epsilon$, e.g.
$\displaystyle \frac{D \epsilon}{Dt} = ...$

I assumed to just substitute for $\displaystyle \phi$ into the transport equation and then expand it out using the product rule but in getting a term as $\displaystyle \frac{D \epsilon}{Dt}$ I end up with terms like
$\displaystyle \frac{D \epsilon^{n-1}}{Dt}$ and $\displaystyle \frac{D k^m}{Dt}$.

Anyone got any ideas how I'd get rid of these unwanted total derivatives?

Thanks,

James
• May 21st 2009, 03:27 PM
the_doc
If I've understood your problem then, by taking logs to base e and then differentiating, you should get:

$\displaystyle \frac{1}{\phi} \, \frac{D \phi}{D t} = \frac{m}{k} \, \frac{D k}{D t} +\frac{n}{\epsilon} \, \frac{D \epsilon}{D t}$ .

Is that what you got? If not, perhaps this will help.