the surface tension induces a normal pressure on the free surface of the fluid which is proportional to the curvature of the surface. On the other hand, a non- ...
Implementation of surface tension effects in Finite Element calculations of flows in the weld pool H. Amin El Sayed, B. Souloumiac, J.B. Leblond, J.M. Bergheau
Surface tension effects are of primary importance to accurately simulate flows that develop in weld pools. Surface tension acts on the fluid flows through two effects. Firstly, the surface tension induces a normal pressure on the free surface of the fluid which is proportional to the curvature of the surface. On the other hand, a non-uniform temperature field at the surface leads to a non-uniform surface tension which then generates a tangential force on the surface (Marangoni effect). In finite element codes of fluid flows, the effects of surface tension are introduced through the efforts it induces as a boundary condition of the Navier-Stokes equations. If to take the Marangoni effect into account does not pose any particular problem in this approach, the calculation of the normal force, which requires the evaluation of the average curvature of the surface (curvature tensor trace), is much more delicate. This average curvature is generally obtained by taking the divergence of the normal vector to the surface with the shape functions of the finite elements and the nodal values of the vectors obtained by some averaging of the normals to the elements containing the nodes considered. In practice this calculation may contain significant errors that will have a negative impact on the accuracy of the results. But the physics teaches us that the surface tension effects are equivalent to those of a fictitious tight membrane stuck onto the interface between the two media in contact. The figure below illustrates this equivalence in a 2D situation.
Figure 1 – Fictitious membrane separating 2 media Let’s consider a fictitious membrane (length ) separating a medium 1, which we assume to be a fluid and a medium 2, which we assume to be a gas. The interface is supposed to be curved toward the fluid with radius of curvature . Denoting by the surface tension at the curvilinear abscissa , the resulting force exerted by the membrane on the fluid is equal to: with:
where and respectively represent the surface tension and the tangent vector to the membrane at curvilinear abscissa . Recalling that , where is the normal vector to the membrane directed toward the curvature centre, one then obtains:
In this equation, one recognizes the curvature effect in the first term of the right hand side and the Marangoni effect in the second one. Instead of taking the surface tension effects into account through the loads they induce on the fluid, the idea is then to directly impose a surfacic spherical stress state in membrane elements incorporated in the mesh and representing the interface. The stress state to impose at any point of the interface, and consequently in membrane elements, can be written: This stress state is naturally taken into account through the expression:
appearing classically in the weak form of the problem, where the strain rate to the nodal values of velocities and
is the operator linking .
The proposed approach does not require any evaluation of the mean curvature of the surface, which makes it effective and accurate. Moreover, it enables to take both the curvature effect and the Marangoni effect into account in a natural and straightforward way. Two examples are presented to illustrate the proposed method. The first one concerns the small oscillations of a fluid, the upper surface of which is free, submitted to both gravity and surface tension. Only the curvature effect is considered here. The fluid is enclosed in a parallelepipedic box and allowed to move freely along its lateral sides; the problem is in fact 2D in a vertical plane. The results of the proposed approach are compared with the analytic solution provided by Prosperetti [2] (figure 2). The second example is a case of weld pool where the surface tension varies along the interface, giving thus rise to Marangoni’s effect; in contrast, the normal force is eliminated by prescribing a zero normal velocity on the flat interface. The resulting nonuniformity of the surface tension generates, via Marangoni’s effect, a vortex-like motion of the fluid. No analytical solution is available for this problem, but one may compare the solutions obtained in 3D with a 2D-axisymmetric solution obtained by five CFD codes [3] by incorporating surface tension in the standard way (figure 3).
Figure 2 – Time oscillations of a fluid submitted to gravity and surface tension
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Figure 3 – Temperature and velocity field along the radius of the disc [1] J.B. Leblond, H. Amin El Sayed, J.M. Bergheau. On the incorporation of surface tension in finite-element calculations. Comptes Rendus Mécanique, vol. 341, 2013, pp.770-775. [2] A. Prosperetti. Motion of two superposed viscous fluids. Phys. Fluids 24 (1981) 1217–1223. [3] P. Girard, M. Bellet, G. Caillibotte, M. Carin, S. Gounand, F. Mathey, M. Médale. Benchmark for fluid flow in weld pool simulation Two-dimensional transient models for arc welding. Internal report, 2005.
