The subgrid Reynolds stress

%$\widetilde{ {u_i}''{u_j}''}= - \nu_t {\partial \widetilde{u_i} \over \partial x_j}$%

Starting from the unmodelled mometum equation

%${\partial {u_i} \over \partial t} + {u_j} { \partial {u_i} \over \partial x_j} = - {1 \over \rho} {d p \over d x_i} + {\partial \over \partial x_j} \left( \nu { \partial u_i \over \partial x_j} \right)$%

##### The averaged momentum equation

we can look at 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

##### The resolved momentum equation

and we can also look at 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.

##### The averaged resolved momentum equation

The interesting thing to do is to look at what the averaged resolved momentum equation equation looks like.

This is all very well in theory but it is not terribly clear how to obtain terms such as %$\widetilde{\overline{{u_i}'{u_j}'}}$%.

##### Modelling the momentum equation

By relooking at the modelling of the resolved mometum equation it becomes a little clearer how these things can be modelled

%${\partial \widetilde{u_i} \over \partial t} + \widetilde {u_j} { \partial \widetilde {u_i} \over \partial x_j} = - {1 \over \rho} {d \widetilde{p} \over d x_i} + {\partial \over \partial x_j} \left( \nu { \partial \widetilde{u_i} \over \partial x_j} + \nu_t {\partial \widetilde{u_i} \over \partial x_j} \right)$%

the averaged resolved equation now becomes %${\partial \overline{\widetilde{u_i}} \over \partial t} + \overline{\widetilde{u_j}} { \partial \overline{\widetilde{u_i}} \over \partial x_j} = -{1 \over \rho} {d \overline{\widetilde{p}} \over d x_i} + {\partial \over \partial x_j} \left( \nu { \partial \overline{\widetilde{u_i}} \over \partial x_j} - \left[ \overline{\nu_t} {\partial \overline{\widetilde{u_i}} \over \partial x_j} + \overline{{\nu_t}' {\partial \widetilde{{u_i}'} \over \partial x_j}} \right] - \overline{\widetilde{{u_i}'} \widetilde{{u_j}'}} \right)$%

it should be noted that %$\nu_t$% is a function of the velocity (e.g. the Smagorinsky model.) so strictly speaking should incorporate the averaging process associated with its internal velocities too.

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Topic revision: r5 - 2008-12-19 - 10:49:57 - JuanUribe
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23 Aug 2019

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