We present enhanced physics-based finite element schemes for two families of turbulence models, the

Energy and helicity play fundamental physical roles in the development of turbulent flow. They are conserved in inviscid flow, are delicately balanced when viscous and body forces are present [

Unfortunately, often one or both of these quantities is not accurately treated in most models (in their continuous form), and thus cannot be correctly handled in their discretizations. Only a very few models correctly balance both energy and helicity, the most prominent being the Navier-Stokes equations (NSE, the true physical model), NS-

A solution to this problem was first introduced in 2004 by Liu and Wang in [

Given a domain

NS-

Recall the balances of energy and helicity for NS-

Given initial condition

While the energy balance of NS-

Balances for NS-

The main obstacles in extending the scheme of [

Denote the zero mean periodic subspaces of

Define the discrete curl operator to be the

Note it is obvious that the linear operator

Since the Laplacian operator applied to a divergence free function

Define the discrete Laplacian

We define the discrete filter

Given an endtime

That solutions exist for Algorithm

We now prove the balances of energy and helicity by the scheme. Note the energy balance is, in a sense, a stability estimate.

The scheme of Algorithm

The energy and helicity balances of the theorem are discrete analogs to the continuous NS-

The energy balance follows from choosing

For the helicity balance, begin by choosing

From the proof, the need for the intricate design of the enhanced-physics scheme becomes more clear. Without the use of the discrete curl in the scheme, a test function could not have been chosen to vanish the nonlinearity in the helicity balance, which in turn would cause the nonphysical creation and dissipation of helicity instead of cascading it from the large to small scales. The derivation of differences in helicities at successive timesteps in (

The development of the discrete curl, Laplacian and filter operators allows for an easy extension to the deconvolution family of models (2.4)–(

Given an endtime

As in the case of no deconvolution, proofs of solution existence and convergence follow similarly to work in [

We now provide the energy and helicity balances for this algorithm.

The scheme of Algorithm

The energy balance follows exactly as in the case without deconvolution. The helicity balance follows similar to the case of NS-

The Stolz-Adams ADM family take a similar mathematical form to the NS-

Technical arguments in [

Though perhaps not obvious, the discrete operators which permitted the development of an enhanced-physics scheme for the NS-

Given an endtime

Since by definition the discrete curl is discretely-divergence free, the discrete Laplacian must be also. Hence filtered elements in

Similar to the NS-

The scheme of Algorithm

For the energy balance, choose

For the helicity balance, choose

We have developed enhanced-physics based finite element schemes for the NS-