In so far as explicit LES is concerned, there are no practical obstacles to combine it with AMR. Say, set the filter length to the local size of the grid and the model will run stably. There are, however, theoretical (and thus accuracy) issues related to commutativity of filtering and differencing operations. For discussion see Marsden,  Vasilyev, and Moin, Construction of commutative filters for LES on unstructured meshes, J. Comput. Phys. 175 (2002) 584–603, and references therein.

Piotr

using implicit-filtering LES you link the size of the grid to the local grid cut-off filter but the real shape of the transfer function you have depends strongly on the adopted scheme.

Using an integral FV-based formulation you can define a local top-hat filtering.

Be careful that the filter length will be local if you use a dynamic procedure where an explicit test-filtering must be implemented.

Keep in mind that the adaptive mesh may not be very effective. Your simulation will need high resolution wherever there is turbulence in order to resolve the local motion of the fluid. You may end up with a situation where your entire grid is fully refined. If that happens the adaptive mesh routine will start to cost more computation time than it gains.

This is a very interesting question and it could have a huge impact on the ability to perform LES of complex high-Re flows. There are 2 main issues to consider:

1. as Piotr explained, if using an explicit LES model, commutation errors are introduced. This is the case regardless of whether filtering is an implicit or explicit operation. The errors will pollute the results near the grid scale. They can be controlled by using the methods pioneered by Vasilyev et al. or by avoiding explicit LES altogether, e.g. using ILES or VMS.

2. Your question is, what should we use as an error estimate for driving AMR in LES? This is far from being a well-resolved question (excuse the pun). Estimates of when an LES is well-resolved are notoriously unreliable; see Davidson, 2009 "Large Eddy Simulations: how to evaluate resolution". A fairly straightforward approach is to use Pope's guideline that 80% of the total turbulent kinetic energy (TKE) should be resolved. Of course, the subgrid TKE is not known and must be modelled by a local estimate in each cell. Knopp et al., 2010 "Enhancement of an industrial finite-volume code for large-eddy-type simulation of incompressible high Reynolds number flow using near-wall modelling" includes a brief discussion of this approach and shows some results. In my paper (Bull et al., 2012 "A finite element LES methodology for anisotropic inhomogeneous meshes") I adopted a more heuristic approach: the error estimate was in terms of 2nd derivatives of the instantaneous velocity, and the mesh was prevented from becoming too fine by imposing limits on maximum number of cells, edge lengths and absolute error bounds. This can be used with ILES and LES. Of course, some tuning of the AMR parameters is required for each particular case, which is not ideal.

As a final note, Allard pointed out that AMR may be too costly in a turbulent flow. There is no obvious way to determine how frequently (in timesteps or flow time) to adapt; some experimentation may be necessary. Of course, a prerequisite is to have a very efficient AMR algorithm.

well, commutation error exists only if you want to commute filter and derivatives... if you do not do that, just use the term G* [Div (vv)] as it is.

SInce some years ago I developed the FV formulation based on such decomposition and the equations contain no commutation errors. Details are in some papers.

Finally, if an implicit filtering is used, I expect that the LES solution will tend to a DNS when the grid is fully refined (by AMR or something else)

Jonathan correctly mentioned that "some" TKE remains within the cell and remains unknown so far, with or without AMR. This is a matter of turbulence closure and simply tuned algebraic expressions have possibly no sufficient physical basis. I attach a paper where this topic is discussed in more general terms providing an opportunity to be applied to the LES-ARM circle of problems. If somebody will try it, I would try to help as far as I can. However, most people look only for solutions for quick applications so that progress in this area is slower than possible.