Variational formulation of three‐dimensional viscous free‐surface flows: Natural imposition of surface tension boundary conditions

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace-Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a case where the exact solution is known (the catenoid). In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test case we achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms.

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“…This term includes both normal and tangential contributions. In [20] it was shown that this integral can be expressed as…”

Section: Relation To Free Surface Flowmentioning

confidence: 99%

High order numerical approximation of minimal surfaces

Tråsdahl¹,

Rønquist²

2011

Journal of Computational Physics

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“…In film problems we set V u = H 2 1 (Ω) and V b = H 1 (Ω), so that the no-slip boundary conditions (2.8) are enforced strongly, whereas the stress and insulating boundary conditions, (2.6), (2.7), and (2.9), must be imposed in a natural (weak) sense (e.g. [15,46]). Taking the free-surface amplitude into account, the full solution space is therefore V = V u × V b × C, which we equip with the direct-sum inner product (…”

Section: Weak Formulationmentioning

confidence: 99%

A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

Giannakis

Fischer

Rosner

2009

Journal of Computational Physics

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We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld (OS) and induction equations for parallel, incompressible MHD in free-surface and fixed-boundary geometries. The schemes' discrete bases consist of Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions. The orthogonality properties of the basis polynomials solve the matrixcoefficient growth problem, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as p = 3,000 with p-independent roundoff error. Accuracy is limited instead by roundoff sensitivity due to non-normality of the stability operators at large hydrodynamic and/or magnetic Reynolds numbers (Re, Rm 4 × 10 4 ). In problems with Hartmann velocity and magnetic-field profiles we employ suitable Gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error. An alternative approach, which involves approximating the forms by means of Legendre-Gauss-Lobatto (LGL) quadrature at the 2p − 1 precision level, is found to yield equal eigenvalues within roundoff error. As a consistency check, we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller than 10 −5 for the least stable mode in free-surface flow at Re = 3 × 10 4 . Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10 −6 . The critical Reynolds number in free-surface MHD is found to be sensitive to the magnetic Prandtl number Pm, even at the Pm = O(10 −5 ) regime of liquid metals.

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“…The key idea to effectively represent this term, first put forth by Ruschak [29] and later generalized to three-dimensional cases by Patera and Ho [21,22], is to employ a weak-form representation of the left-hand side of Equation (16), so that the mean curvature is expressed in terms of first derivatives of surface shape. We follow the ideas of Ruschak [29] and Patera and Ho [21,22] by invoking an identity from differential surface geometry [31] to express the first curvature s of the surface in terms of first-order derivatives. Here, we again show only the x-component of the derivation for brevity…”

Section: Finite Element Formulationmentioning

confidence: 99%

An assessment of a parallel, finite element method for three‐dimensional, moving‐boundary flows driven by capillarity for simulation of viscous sintering

Zhou

Derby

2001

Numerical Methods in Fluids

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SUMMARYA parallel, finite element method is presented for the computation of three-dimensional, free-surface flows where surface tension effects are significant. The method employs an unstructured tetrahedral mesh, a front-tracking arbitrary Lagrangian-Eulerian formulation, and fully implicit time integration. Interior mesh motion is accomplished via pseudo-solid mesh deformation. Surface tension effects are incorporated directly into the momentum equation boundary conditions using surface identities that circumvent the need to compute second derivatives of the surface shape, resulting in a robust representation of capillary phenomena. Sample results are shown for the viscous sintering of glassy ceramic particles. The most serious performance issue is error arising from mesh distortion when boundary motion is significant. This effect can be severe enough to stop the calculations; some simple strategies for improving performance are tested.

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Variational formulation of three‐dimensional viscous free‐surface flows: Natural imposition of surface tension boundary conditions

Cited by 37 publications

References 9 publications

High order numerical approximation of minimal surfaces

High order numerical approximation of minimal surfaces

A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

An assessment of a parallel, finite element method for three‐dimensional, moving‐boundary flows driven by capillarity for simulation of viscous sintering

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