This can be resolved by insisting that be minimized subject to the constraint . This leads to an equation for that is nonlinear Here the suffix + indicates the "positive part of" that is, . Even though this superficially looks like "clipping" it is not an ad hoc scheme but a bonafide solution of the constrained variational problem. This DLM(+) model was found to be stable and yielded excellent results for forced and decaying isotropic turbulence, channel flows and a variety of other more complex geometries. If a flow happens to have homogeneous directions (let us say the directions x and z) then one can introduce the ansatz . The variational approach then immediately yields Lilly's result with averaging over homogeneous directions without any need for ad hoc modifications of a prior result. One shortcoming of the DLM(+) model was that it did not describe backscatter which is known to be a real "thing" from analyzing DNS data. Two approaches were developed to address this. In one approach due to Carati et al. a fluctuating force with amplitude determined by the fluctuation-dissipation theorem is added in analogy to Landau's theory of fluctuating hydrodynamics. In the second approach, one notes that any "backscattered" energy appears in the resolved scales only at the expense of energy in the subgrid scales. The DLM can be modified in a simple way to take into account this physical fact so as to allow for backscatter while being inherently stable. This k-equation version of the DLM, DLM(k) replaces in the Smagorinsky eddy viscosity model by as an appropriate velocity scale. The procedure for determining remains identical to the "unconstrained" version except that the tensors , where the sub-test scale kinetic energy K is related to the subgrid scale kinetic energy k by (follows by taking the trace of the Germano identity). To determine k we now use a transport equation where is the kinematic viscosity and are positive coefficients representing kinetic energy dissipation and diffusion respectively. These can be determined following the dynamic procedure with constrained minimization as in DLM(+). This approach, though more expensive to implement than the DLM(+) was found to be stable and resulted in good agreement with experimental data for a variety of flows tested. Furthermore, it is mathematically impossible for the DLM(k) to result in an unstable computation as the sum of the large scale and SGS energies is non-increasing by