## Simulation of Three-Dimensional Bénard-Marangoni Flows Including Deformed Surfaces (2009)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Bjøntegaard09simulationof,

author = {Tormod Bjøntegaard and Einar M. Rønquist},

title = {Simulation of Three-Dimensional Bénard-Marangoni Flows Including Deformed Surfaces},

year = {2009}

}

### OpenURL

### Abstract

We present a coupled thermal-fluid model for Bénard-Marangoni convection in a three-dimensional fluid layer. The governing equations are derived in detail for two reasons: first, we do not assume a flat free surface as commonly done; and second, we prepare for the use of flexible discretizations. The governing equations are discretized using spectral elements in space and an operator splitting approach in time. Since we are here primarily interested in steady state solutions, the focus is on the spatial discretization. The overall computational approach is very attractive to use for several reasons: (i) the solution can be expected to have a high degree of regularity, and rapid convergence can be expected; (ii) the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerful results from differential geometry to easily be exploited; (iii) free surface deformation can readily be included; (iv) both normal and tangential stresses are conveniently accounted for through a single surface integral; (v) no differentiation of the surface tension is necessary in order to include thermocapillary effects (due to integration-byparts twice); (vi) the geometry representation of the free surface need only be C 0 across element boundaries even though curvature effects are included. Three-dimensional simulation results are presented, including the free surface deflection due to buoyancy and thermocapillary effects.

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Citation Context ...y orthogonal (gα·g β = δαβ). The free surface boundary conditions along a surface patch Γ can then be expressed278 T. Bjøntegaard and E. M. Rønquist / Commun. Comput. Phys., 5 (2009), pp. 273-295 as =-=[15, 16]-=- niσijnj = γκ, (2.13) ) . (2.14) tiσijnj= ti tiσijnj = −ti ( g α i Here, κ is twice the mean curvature, ni, i = 1,2,3, is the outward unit normal on the surface, and ti, i = 1,2,3, represents any tang... |

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Citation Context ...ity can be exploited when considering the subsequent numerical treatment. There is also significant flexibility in terms of the extension of the domain velocity from the boundary to the interior; see =-=[10]-=-. Finally, we need to impose initial conditions for the temperature and the velocity. 5 Discretization Our starting point for the numerical discretization is the ALE formulation presented above. The d... |

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Citation Context ...ayleigh’s [22] subsequent stability analysis also assumed buoyancy-driven convection. Only several decades later were Bénard experiments correctly interpreted through surface-tension-gradient effects =-=[4, 21, 23]-=-. To our knowledge, the deflection of the free surface has never before been investigated using simulation tools; only a fixed and flat ”free surface” has been used in earlier numerical studies [19,24... |

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Citation Context ...ntageous to express these conditions using surface-intrinsic coordinates. However, before giving the expressions, we first introduce some necessary notation and definitions from differential geometry =-=[9, 15, 26]-=-. r 2 Γ g2 g1 φΓ r 1 Figure 2: A plot depicting part of a deformed surface Γ in three dimensions. We assume that there exist a parameterization of this surface patch using two independent coordinates.... |

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Citation Context ...1, 23]. To our knowledge, the deflection of the free surface has never before been investigated using simulation tools; only a fixed and flat ”free surface” has been used in earlier numerical studies =-=[19,24]-=-. We assume that this is partially due to the fact that the free surface deformation is small, but also partially because the imposition of the simultaneous curvature and surface-gradient effects alon... |

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Citation Context ...1, 23]. To our knowledge, the deflection of the free surface has never before been investigated using simulation tools; only a fixed and flat ”free surface” has been used in earlier numerical studies =-=[19,24]-=-. We assume that this is partially due to the fact that the free surface deformation is small, but also partially because the imposition of the simultaneous curvature and surface-gradient effects alon... |

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Citation Context ... faces, with each face representing a surface patch Γ as discussed earlier; see Fig. 2. The spectral element method has previously been used to simulate three-dimensional, viscous, free surface flows =-=[5, 12]-=-, however, the more general case involving BénardMarangoni flows with deformable free surfaces has not been considered before using any computational method. In the rest of this section we thus focus ... |

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Citation Context ...Θ)κni+Ma ∂Ωγ(t) ( g α i Ω(t) vi ∂σ ij ∂x j dΩ. ∂ (1−Θ))) dS (3.5) ∂rα The curvature-normal product κni can again be replaced with a very powerful expression presented in [12] and derived in detail in =-=[3]-=-, κni = 1 ∂(gg g α i ) ∂rα , (3.6)282 T. Bjøntegaard and E. M. Rønquist / Commun. Comput. Phys., 5 (2009), pp. 273-295 where g is the square root of the determinant of the metric tensor; see (2.12). ... |

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Citation Context ...ture (i.e., thermocapillary forces), or both effects can be present at the same time. However, the free surface deformation associated with these cells has previously only been studied experimentally =-=[2, 6, 14]-=- or analyzed analytically using linear stability analysis [23]. It is known experimentally that the free surface will either be depressed or elevated over each hexagonal cell, depending on whether sur... |