A Numerical Description of a Liquid-Vapor Interface Based on the Second Gradient Theory

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Abstract
The direct numerical simulation of liquid-vapor two-phase flows can be a useful tool both to investigate complex physical problems and provide closure relations used in models based on averaged equations. Several numerical methods that can be applied to liquid-gas problems involving immiscible fluids are able to track interfaces that are likely to merge or separate. These methods are based on an artificial enlargement of the interface, which prevents numerical oscillations, and they actually use a mixing model for the fluid behavior within the interfacial zone thus created. The main advantage of these methods is that only one set of equations has to be solved to obtain the flow field in each phase and to track the interfaces naturally. Although adaptation of these methods to phase-change problems is difficult, artificial enlargement of the interface can still be maintained if an effort is made to model the fluid within the interfacial zone. The second gradient theory is particularly adequate for that purpose since it allows a single set of equations of motion and energy to be written for the whole system, i.e. the liquid and vapor phases as well as the interfacial zone. Unusual terms then naturally appear in these equations and correspond to the transformation of the surface tension into a volumetric property. This article shows that it is possible to increase artificially the thickness of the interface without changing the values of the surface tension or the heat of vaporization, provided that ad hoc thermodynamic closure relations are determined. An example of the application of the second gradient theory is given for an isothermal system near the critical point at which the liquid-vapor interface is physically macroscopic. A numerical investigation shows that the interfacial zone keeps its macroscopic thickness as well as a strong cohesion even when its thickness is perturbed.

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