Hello,

I'm having a problem with appropriate discretization of the subgrid force in my incompressible LES NS-equations. I'm using higher order explicit central finite differences (2nd->10th order).

Suppose I obtained the subgrid stresses TAUij=-2*nu_t*Sij with nu_t smagorinsky viscosity, with the above mentionned difference schemes. To obtain the subgrid force, I have to take the divergence of this stress or in other words, I have to calculate the first order derivatives of this stress. If I do this straightforwardly I obtain a non-compact stencil for the subgrid force which does not "see" spurious modes.

My question is, how do one obtain a compact discretization in explicit higher order central FD on a regular mesh?

Should I discretize it like: dTAUdx=-2*(dnu_tdx*Sij + nu_t*d2udx2) to obtain compact operators?

Or should I use interpolation? If yes, how do you do this in higher order discretizations?

For 2nd order FD twice applying central differences leads to stencil 1 0 -2 0 1 wich can be solved using interpolation such that one obtains for second order derivative 1 -2 1.

For fourth order FD twice applying central difference stencil [1 -8 0 8 -1]/12dx gives not the wide version of the second order derivative for 4th order accuracy, but something completely else... How to interpret this?

What about 6th 8th etc?

Is it important for SGS force to be compactly discretized or doesn't it matter as long as molecular viscosity is compactly discretized such that this one damps spurious modes?

Thanks

Dieter

I don't know if this will help (since they don't use subgrid scale models) but Visbal et al have done some LES modeling using compact 6th order schemes with high order filtering. They have found that just using the filter without the subgrid model gives good results...once again this doesn't answer your question directly but it might help.

High order Central difference schemes do not have built in dissipation and also support non physical waves and hence can become unstable if spurious waves are not filtered(due to boundary conditon,grid uniformity).Hence there is a need for filtering for such applications.An important thing to consider in filtering is a wavenumber analysis of the modes because filtering might remove physical waves too.The work of visbal employs a sixth order compact scheme with 10th order filtering and they have provided many benchmark validations to show that the effect of filtering on the physical solution can be kept minimum by controlling the filtering.another attractive option is to use upwinding.but depending on the application the dissipation might be pretty high.Also some of the other options are optimized DRP scheme with selective filtering ( Tam ), Optimized prefactored scheme with filtering ( Hixon) and many similar schemes which can be found in the workshops on computational aeroacoustics.