Free surfaces boundaries

Nichols, B. D., Hirt, C. W. and Hitchkiss, R. S., 1980. SOLA-VOF a solution algorithm for transient fluid flow with multiple free surface boundaries. Los Alamos Scientific Laboratories Report No. La-8355, Los Alamos, NM. 

As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the model equations. The developed algorithm for the solution of the described model equations and updating of the free surface boundaries is as follows ... 

Step 2 an initial configuration representing the partially filled discretized domain is considered and an array consisting of the appropriate values of F - 1, 0.5 and 0 for nodes containing fluid, free surface boundary and air, respectively, is prepared. The sets of initial values for the nodal velocity, pressure and temperature fields in the solution domain are assumed and stored as input arrays. An array containing the boundary conditions along the external boundaries of the solution domain is prepared and stored. 

Initial distribution and the predicted free surface boundaries within the twin-blade mixer represented by the mesh configurations shown in Figure 5.4, after 30, 60 and 90° rotation of the left blade are presented in Figures 5.5a to 5.5d, respectively. Samples of the predicted velocity fields after 30 and 45 ° rotation of the left rotor are shown in Figures 5.6a to 5.6b, respectively. The finite element... 

The velocity potential satisfies the Laplace equation and the free surface boundary condition on cf> is obtained by differentiating Eq. (10.4.14) with respect to t and eliminating d ldt from Eq. (10.4.16). For waves on deep water the second boundary condition on the potential is supplied by the requirement that there are no disturbances deep in the water, or that = constant as... 

The observed stabilizing effect of surfactants toward convection induced by surface tension has been confirmed theoretically in a recent paper by Berg and Acrivos (B13), in which the stability analysis technique and the physical model were the same as Pearson s except that the free-surface boundary condition [(iii) of Table III] took into account the presence of surface active agents. Critical values for the Thompson number were computed as functions of two dimensionless parameters, one embodying the surface viscosity and the other the surface elasticity. ... 

Figure 18. Surface coverage in units of the complete -s/s monolayer for N2 on graphite near the melting transition from Monte Carlo simulations. Triangles 256 molecules forming an isolated patch subject to free surface boundary conditions. Circles (squares) 16 (64) molecules in the NPTensemble with deformable periodic boundary conditions. Inverted triangles same as circles but obtained from a cooling run starting in the fluid phase. (Adapted from Fig. 1 of Ref. 301.)...

In ceramic extrusion we typically have to deal with inflow- and outflow boundaries, (moving) walls and free surface boundaries for which the appropriate mathematical formulation will be given in the subsequent paragraphs. Since we use pressure, temperature and velocity as our independent set of variables, the boundary conditions must be expressed in terms of p, T, V. They can take two different forms The dependent variables are specified along the boundary (Dirichlet boundary condition) or the directional derivatives of the dependent variables are prescribed Neumann boundary condition). 

Free surfaces are often encountered in extmsion processes, e.g. when the shape of the extrudate leaving the die is of interest. UnUke the boundaries discussed so far, the shape of a free surface boundary is not know a priori since it evolves as part of the solution. Thus two boundary conditions are necessary a kinematic condition that signifies that the particles at the free surface move with the local fluid velocity and a dynamic condition that assures force balance at the surface. The kinematic condition is expressed as... 

The first spray phenomenon that needs to be modeled is the atomization process, that is, the disintegration of the bulk liquid into tiny droplets. The atomization process can be separated into inner-nozzle and outer-nozzle effects. The forces that govern the inner-nozzle atomization include cavitation-induced and turbulence-induced disturbances of the liquid. Once the liquid exits the nozzle, it interacts with the gaseous environment that induces disturbances on the liquid-gas interface caused by aerodynamic and inertial forces. Also, when the liquid exits the nozzle, it experiences a discontinuity in the boundary condition, namely, from the fixed boundary of the nozzle orifice to a free surface boundary. This abrupt change in the boundary condition leads to disturbances of the liquid that influence the atomization process. In general, the atomization of a bulk liquid is a very complex process and is still the subject of intensive research. 

C.W. Hirt, B.D. Nichols Volume of fluid method for the dynamics of free surface boundaries, J. Comput. Phys., 1981, 39, 323-345. 

Following the fluid flow analogy, the most obvious choice for a free surface boundary condition is to allow the shear stresses to vanish and thereby force the shear rate to go to zero (i.e., du/dy = 0). Although this condition has been used in chute flow calculations (Savage, 1979 Campbell and Brenen, 1985 etc.) it forces the velocity profile to a... 

Combined Kinematic and D mamic Free Surface Boundary Condition (CKDFSBC) ... 

Domain mapping of the WMBVP follows the theory by Joseph. The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank Nicholson (C N) method. s each iteration of the C-N method, the potential field is computed by the conjugate gradient method. The wavemaker motion E y/h,t) is assumed to be periodic with period T — 2tt/u , and the WMBVP with the surface tension f is given by... 

The velocity potential ip is the field variable, and the free surface y may be expressed in terms of p> by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.h The velocity potential ip and the free surface displacement r] are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following ... 

Assuming incompressible fluid and irrotational flow motion, the velocity potential exists, which satisfies the Laplace equation. Linearizing the free-surface boundary conditions, the following boundary value problem for the velocity potential x,z,t) is obtained ... 

Linearizing the free-surface boundary conditions, the following boundary value... 

Irregular (nonplanar) free-surface boundary conditions are far more difficult to implement in FD methods. Typically, such methods require a much finer sampling of the wave field for accurate results (e.g., Robertsson 1996 Ohminato and Chouet 1997). 

The solution domain is shown in Figure 2. Velocity is fully specified at the inflow and outflow boundaries, Fi and Fo, respectively. Symmetry boundary conditions are imposed along the centerline, labeled Fj. Free surface boundary conditions are imposed on the edge of the domain, denoted Ff. The specific boundary conditions are as follows. 

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