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Thread: Total derivative manipulation

  1. #1
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    Apr 2009
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    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
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  2. #2
    Member
    Joined
    May 2009
    Posts
    127
    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.
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