Interface capturing

Among interface capturing methods, one of the most popular and most successful schemes is the volume-of-fluid (VOF) method dating back to the work of Hirt and Nichols [174]. The VOF method is based on a volume-fraction field c, assuming values between 0 and 1. A value of c = 1 indicates cells that are filled with phase 1, and phase 2 corresponds to c= 0. Intermediate values of c indicate the position of the interface between the phases however, the goal is to maintain a sharp interface in order to identify the different fluid phases uniquely. Volumes assigned to the different phases are moving with the local flow velocity W , and therefore the evolution of c is determined by a convection equation ... 

Yabe, T. Interface Capturing and Universal Solution of Solid, Liquid and Gas by CIP Method . Proceedings of the High-Performance Computing of Multi-Phase Flow, Tokyo, July 18-19, 1997. 

Caylor et al. [134] performed ellipsometric measurements at the liquid-liquid interface of Pitzer s system. The experiments yielded a decrease in ellipticity, when approaching the critical point. This is in clear contrast to a (/ — v) divergence predicted theoretically [135] and observed experimentally [136] for nonionic Ising mixtures. The results point toward an anisotropic interface caused by an orientation of ion pairs perpendicular to the interface. A rough model for such an interface captured some of the observed features. 

Chang, Y.C., Hou, T.Y., Merriman, B., and Osher, S. (1996) A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, Journal of Computational Physics, Vol. 124, pp. 449-464. 

Chang YC, Hou TY, Merriman B, Osher S (1996) A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows. J Comput Phys 124 449-464. 

Fig. 17.1 Different interfacial flow solution methodologies (a) interface fitting, (b) interface capturing, and (c) interface tracking [4] (Recopied from [4] with permission)...

The single-fluid approach has been undertaken utilizing Eulerian methods such as VOF, LS, and the mixed Eulerian-Lagrangian methods [11, 12], For most interface capturing schemes, which use single-fluid formulation, an additional equation is solved to obtain the interface evolution and topology. This equation governs the advection of a variable that can be attributed to the interface. The equation of interface motion is... 

The scalar x is defined differently in different interface capturing techniques. For example, x is volume fraction in the VOF volume tracking method, a distance function in the LS method, or a Heaviside (step) function in the front tracking method. 

Interface Capturing Schemes for Free-Surface Flows... 

Broadly speaking, there are two computational approaches to solving fluid flow problems that include an evolving interface between two fluid phases. These can be categorized as interface tracking and interface capturing. The former refers to methods that explicitly parameterize the interface and follow the points on the surface... 

In this contribution, first a number of fundamental concepts that are central to interface capturing are presented, including definitions of level set functions and unit normal and curvature at an interface. This is followed by consideration of kinematic and dynamic boundary conditions at a sharp interface separating two immiscible fluids and various ways of incorporating those conditions into a continuum, whole-domain formulation of the equations of motion. Next, the volume-of-fluid (VOE) and level set methods are presented, followed by a brief outlook on future directions of research and other interface capturing/tracking methods such as the diffuse interface model and front tracking. 

Interface Capturing Schemes for Free-Surface Flows, Fig. 1 A circle represented by a signed distance function (left panel) and with a phase-field function (right panel)... 

In contrast, in interface capturing methods, it has been found conveitient to formulate a whole-domain [6] or one-fluid [7] model that automatically captures the appropriate conditions at the interface between the fluids. The momentum equation is this formulation can be written as... 

Numerical interface capturing methods consist of various techniques for integrating the above system of conservation of mass and momentum equations, together with advection of an appropriate level set or phase-indicator function, to enable an approximate localization of the interface and proper assignation of fluid properties. We now describe two widely used methods to accomplish this volume-of-fluid and level set methods. 

The level set method [1,2] is an interface capturing scheme in which the evolution of an interface is tracked by evolving a level set function (f>(x, t) throughout space but focusing on the location of a specific level surface (or curve in 2D) of (f> to capture the position of the desired interface. In contexts other than two-phase fluid flow, the evolution equation for the level set function can account for advection by a specified external velocity field v(x, t), as well as propagation of the interface in the direction of the local normal vector at a constant speed a, and the motion of the interface in the normal direction at a speed proportional to the local curvature k with proportionality constant —b. The resulting evolution equation for cj) then takes the form... 

There are a number of limitations in interface capturing schemes that continue to be addressed by researchers. One is the problem of so-called parasitic or spurious currents [6] which can be attributed to the interfacial regions being underresolved. These currents can be reduced via formulations that strictly enforce energy conservation and account for exchanges of kinetic and surface energies [11]. There are also difficulties in computations at small values of surface tension when the computed objects (drops, filaments, bubbles) become very small and cannot be... 

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