A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

Abstract

In this paper, we formulate a level set method in the framework of ÿnite elements-semi-Lagrangian methods to compute the solution of the incompressible Navier-Stokes equations with free surface. In our formulation, we use a quasi-monotone semi-Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier-Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and exible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the ow past a NACA0012 foil at high Reynolds number.A SEMI-LAGRANGIAN LEVEL SET METHOD 1113 mesh [15], our quasi-monotone interpolation is very general in the sense that it is suitable for any type of mesh without any additional e ort. We should understand quasi-monotone interpolation as positive interpolation or, better perhaps, as a type of essentially non oscillatory interpolation because it can hold oscillations with amplitude O(h p+1 ), where h denotes the mesh size parameter and p the degree of the polynomial interpolation. Moreover, our method to provide quasi-monotonicity is not speciÿcally designed for Lagrange interpolation; on the contrary, it can be executed as well with Hermite interpolation, spline interpolation or even trigonometric interpolation. (ii) We do reinitialization (redistancing) of the level set function by using Sussman and Fatemi algorithm [16] in our semi-Lagrangian framework. The numerical solution of the Navier-Stokes equations is calculated as proposed by Allievi and Bermejo in Reference [12], using the quasi-monotone extension of the conventional semi-Lagrangian interpolation ÿrst introduced in Reference [14].

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

“…Eq. (12), which is provided here only for an easier representation of the orders of the truncation errors. To demonstrate how the proposed method can be applied to non-uniform meshes, this paper investigated its performance for both a 1-D and a 3-D stretched mesh cases in later sections.…”

“…Since it was first proposed by Courant et al [1], the semi-Lagrangian method has been widely used to solve advections of the Navier-Stokes equation for various problems including weather forecasting, oceanic flows [2][3][4], shallow water flows [5,6], building airflows [7][8][9], multiphase simulations [10][11][12], and fluid flow simulations for game engines [13,14]. The major benefit of using the semi-Lagrangian method is its unconditional stability even for large time steps, thus offering faster solution than other computational fluid dynamics solvers.…”

Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid

“…we see that ξ = (Id + dη) −1 − Id will be small. If we define the operator P (ξ, u, q, φ, σ) as the left-hand side of (14) for equations (14) 1 to (14) 5 and the initial conditions (14) 7 to (14) 8 , then (14) amounts to solving P (ξ, u, q, φ, σ) = (0, 0, 0, 0, 0, u 0 , σ 0 ), for u vanishing on S B (see (14) 9 ) and Φ(t = 0) = 0 (see (14) 6 ). The orders of magnitude of various terms must now be identified so as to see (14) as a perturbation of an invertible system.…”

“…The orders of magnitude of various terms must now be identified so as to see (14) as a perturbation of an invertible system. In system (14), there are some source terms : gravity and the initial curvature that are not small even for small times. They are of zeroth order and will be put in P (0, 0, 0, 0, 0).…”

“…Proceeding as in [4], we collect the remaining terms in a new operator acting on (u, q, φ, σ) with the zeroth order in ξ (we put ξ in E). Moreover, we keep only the linear in Φ operator from (14) and put the h.o.t. in Φ terms in E 5 (since Φ is initially vanishing these nonlinear terms will be small in small time).…”