Abstract
We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided.
Highlights
The vorticity fields in the rotational form of the nonlinear convective term, see (2d), serve as a means of exchanging information between the two evolution equations. This leads to a leap-frog like scheme that handles the nonlinear rotational term by staggering in time the velocity and vorticity such that the resulting discrete algebraic systems are linearized and decoupled. The objective of this novel approach is the construction of a discretization which conserves mass, kinetic energy and helicity for the incompressible Navier-Stokes equations in the absence of dissipative terms and predicts the proper decay rate of kinetic energy and helicity based on the global enstrophy and an integral quantity of vorticity, respectively
Re = 100 are presented in Fig. 3 where the results shown in the top diagrams verify the dissipation rate of kinetic energy derived in (30) and (31) and the results in the bottom-left diagram are in agreement with the dissipation rate of helicity, see (44)
We introduce a discretization which satisfies pointwise mass conservation and, if in the absence of dissipative terms, conserves total kinetic energy and total helicity and, otherwise, properly captures the dissipation rates of total kinetic energy and total helicity for the 3D incompressible Navier-Stokes equations
Summary
= − (ω × u) · ω − ω · (ω × u) + ∇ × ∇ P · u − P · ∇ · ∇ × u = 0 , where we have used (i) the definition of vorticity ω := ∇ × u, (ii) integration by parts on the second and fourth terms in the right side of the first identity, (iii) the vector calculus relation (5), and (iv) the identities ∇ × ∇ (·) ≡ 0 and ∇ · ∇ × (·) ≡ 0 These conservation laws for kinetic energy (in 2D and 3D), enstrophy (in 2D), and helicity (in 3D), are the expression of a more general structure underlying the incompressible Euler equations: the Hamiltonian structure, [10,11,12,13,14,15,16]. In the same way as energy conserving schemes have shown to substantially contribute to a higher fidelity in simulations, see for example [3,44,45,46,47,48,49], due to the connection between the cascades of energy and helicity, helicity conserving schemes should present a positive impact towards improving the simulation accuracy
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