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Reynolds Stresses in Cylindrical Coordinates

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Old   February 7, 2009, 14:38
Default Reynolds Stresses in Cylindrical Coordinates
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I'm struggling with the derivation of the above. I have the 'usual' stress terms (in cartesian coordiantes they are rho u v, essentially), but I'm thinking that there are extra terms. For any one who 'speaks' latex2e, I've the text at the bottom of this post. anyone able to tell me if I am missing something?



\begin{subequations} \begin{multline} \rho \left(\frac{\pa \overline{u_r}}{\pt} + \overline{u_r} \frac{\pa \overline{u_r}}{\pa r} + \frac{\overline{u_{\theta}}}{r} \frac{\pa \overline{u_r}}{\ptheta} + \overline{u_z} \frac{\pa \overline{u_r}}{\pa z} - \frac{\overline{u_{\theta}}^2}{r}\right) =-\frac{\pa p}{\pa r} +\dots\\ \dots+\mu \left[\frac{1}{r}\frac{\pa}{\pr}\left(r \frac{\pa \overline{u_r}}{\pa r}\right) + \frac{1}{r^2}\frac{\pa^2 \overline{u_r}}{\pa \theta^2} + \frac{\pa^2 \overline{u_r}}{\pa z^2} -\frac{\overline{u_r}}{r^2} - \frac{2}{r^2}\frac{\pa \overline{u_{\theta}}}{\pa \theta}\right] + \rho g_r+\frac{\mu}{3}\left(\frac{1}{r}\frac{\pa(r \overline{u_r})}{\pr}\right)-\dots\\ \dots-\rho\frac{\pa}{\pr}\left(\overline{u'_{r}u'_{r}}\r ight)-\frac{\rho}{r}\frac{\pa}{\ptheta} \left(\overline{u'_{r}u'_{\theta}}\right)-\rho\frac{\pa}{\pz}\left(\overline{u'_{r}u'_{z}}\r ight) - \frac{\overline{u'_{\theta}}^2}{r} \end{multline} \begin{multline} \rho \left(\frac{\pa u_{\theta}}{\pa t} + u_r \frac{\pa u_{\theta}}{\pa r} + \frac{u_{\theta}}{r} \frac{\pa u_{\theta}}{\pa \theta} + u_z \frac{\pa u_{\theta}}{\pa z} + \frac{u_r u_{\theta}}{r}\right) =-\frac{1}{r}\frac{\pa p}{\pa \theta} +\dots\\ \dots+\mu \left[\frac{1}{r}\frac{\pa}{\pa r}\left(r \frac{\pa u_{\theta}}{\pa r}\right) + \frac{1}{r^2}\frac{\pa^2 u_{\theta}}{\pa \theta^2} + \frac{\pa^2 u_{\theta}}{\pa z^2} + \frac{2}{r^2}\frac{\pa u_r}{\pa \theta} - \frac{u_{\theta}}{r^2}\right] + \rho g_{\theta}+\frac{\mu}{3}\left(\frac{1}{r}\frac{\pa (u_{\theta})}{\ptheta}\right)-\dots\\ \dots-\rho\frac{\pa}{\pr}\left(\overline{u'_{r}u'_{\thet a}}\right)-\frac{\rho}{r}\frac{\pa}{\ptheta} \left(\overline{u'_{\theta}u'_{\theta}}\right)-\rho\frac{\pa}{\pz}\left(\overline{u'_{\theta}u'_{ z}}\right) + \frac{\overline{u'_{r}u'_{\theta}}}{r} \end{multline} \begin{multline} \rho \left(\frac{\pa u_z}{\pa t} + u_r \frac{\pa u_z}{\pa r} + \frac{u_{\theta}}{r} \frac{\pa u_z}{\pa \theta} + u_z \frac{\pa u_z}{\pa z}\right) =-\frac{\pa p}{\pa z} + \dots\\ \dots+\mu \left[\frac{1}{r}\frac{\pa}{\pa r}\left(r \frac{\pa u_z}{\pa r}\right) + \frac{1}{r^2}\frac{\pa^2 u_z}{\pa \theta^2} + \frac{\pa^2 u_z}{\pa z^2}\right] + \rho g_z+\frac{\mu}{3}\left(\frac{\pa(u_{z})}{\pz}\righ t)-\dots\\ \dots-\rho\frac{\pa}{\pr}\left(\overline{u'_{r}u'_{z}}\r ight)-\frac{\rho}{r}\frac{\pa}{\ptheta} \left(\overline{u'_{\theta}u'_{z}}\right)-\rho\frac{\pa}{\pz}\left(\overline{u'_{z}u'_{z}}\r ight) \end{multline}\label{navier-stokes-rs} \end{subequations}
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