the averaged resolved momentum equation which combines both:

* the averaged momentum equation where %$u=\overline{u} + u'$% and %$\overline{u'}=0$% with the bar representing spatial and/or time averages and the %$'$% prime fluctuations around the average.

and

* the resolved momentum equation where %$u=\widetilde{ u} + u''$% and %$\widetilde{u''}=0$% with the tilde representing the resolved velocity and the %$''$% double prime the subgrid velocity

looks like

%${\partial \overline{\widetilde{u_i}} \over \partial t} + \overline{\widetilde{u_j}} { \partial \overline{\widetilde{u_i}} \over \partial x_j} = -{1 \over \rho} {d \overline{\widetilde{p}} \over d x_i} + {\partial \over \partial x_j} \left( \nu { \partial \overline{\widetilde{u_i}} \over \partial x_j} - \left[ \underbrace{(\widetilde{\bar{u_i}\bar{u_j}} + \widetilde{\overline{{u_i}'{u_j}'}})}_{\widetilde{ {u_i}{u_j}}} - \underbrace{(\widetilde{\bar{u_i}} \widetilde{\bar{u_j}} + \overline{\widetilde{{u_i}'} \widetilde{{u_j}'}})}_{\widetilde{u_i} \widetilde{u_j}} \right] - \overline{\widetilde{{u_i}'} \widetilde{{u_j}'}} \right)$%

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Topic revision: r2 - 2009-03-04 - 10:02:54 - YacineKahil
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23 Aug 2019

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