Abstract
We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly enforced. However, it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.
Funding statement: The work was partially supported by NSF grant DMS 2011490.
Acknowledgment
ClemsonUniversity is acknowledged for allowing a generous amount of computation time on the Palmetto cluster.
References
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A Momentum/angular momentum conservation of NS-α and Leray-α formulations
Here we show that the NS-α and Leray-α formulations do not conserve momentum or angular momentum if div u ≠ 0 and div w ≠ 0 where w represents the filtered velocity ū. Further note u = Fw where F = −α2ΔI + I.
A.1 NS-α
Recall the nonlinear term of the NS-α formulation is
Test (A.1) with ei for i = 1, 2, 3. After applying the space–time divergence theorem and rearranging some we get
Assuming ν = f = 0, (A.2) simplifies into
If the nonlinear term is equal to zero, then we will have momentum conservation. We now check this:
where the above two equalities come from vector identities. Also note that because ei is a vector of scalars, ((∇ ⋅ ei)w, u) = (w ⋅ ∇ei , u) = 0. This leaves us with
which we cannot conclude is zero, hence we cannot say that the NS-α formulation preserves momentum. For angular momentum, we test (A.1) with φi and the algebra works out similar to momentum,
Since ∇⋅ φi = 0 for i = 1, 2, 3, we have
Also recall using (3.15) in Theorem 3.3, we have
This gives us
Much like with momentum, we cannot conclude that this quantity is zero, and we expect it is not zero.
A.2 Leray-α
Recall the nonlinear term of the Leray-α formulation is
We test (A.3) with ei for i = 1, 2, 3 and integrate. Similar to (A.2)
Assuming ν = f = 0, (A.4) simplifies to
If the nonlinear term is equal to zero, thenwe will have momentum conservation. Using (2.6) on the nonlinear term we get
which is not zero when ∇⋅ w ≠ 0. Hence momentum is not necessarily conserved.
For angular momentum we test (A.3) with φi for i = 1, 2, 3 and it simplifies to
Now similarly to the momentum proof, we have for the nonlinear term
where the first term disappears by applying (3.15) similarly to Theorem 3.3 (and the angular momentum proof for NS-α in Appendix A.1). Thus the nonlinear term does not vanish, so angular momentum is not conserved.
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