Session BK: Free Surface Flows I

A Special Quadrature for Boundary Integral Techniques applied to thin regions.

Boundary integral techniques for elliptic problems contain a kernel that is singular. Special methods are available for a single surface that can treat the principal-value integral accurately. But when there are two surfaces close together, integrals for points on one surface must be conducted along the other, where the kernel is nearly singular and the integrand shows rapid variation in the region of closest approach. A new technique based on analytic continuation removes the source of the difficulty providing high accuracy even when the spacing between surface points is larger than the distance between the surfaces. The application of this new technique to the motion of thin vortex layers undergoing Kelvin-Helmholtz instability and to thin liquid layers undergoing Rayleigh-Taylor instability will be described.   

A simplified linear free-surface treatment for RANS calculations

There are many different techniques used to capture free-surface flows in RANS calculations (e.g. level-set, mesh deformation, VOF). These methods all have significant costs attached to them and while they are useful for problems where there is strong nonlinearity or wave breaking are present, these methods can prove overly costly for simple ship drag problems due to the high number of cells that must be located in the vicinity of the free surface. In general ship drag problems the effect of the free surface on the boundary-layer development and the interaction between the two is the major concern, not the wave breaking effects. This can be seen in the success of classical linear predictions for many of these types of problems. The method to be presented approximates the free surface by solving a simple linear pressure based boundary condition at the mean free surface. This approach is straightforward and requires only minimal modifications to existing RANS codes, while significantly decreasing the meshing requirements over other forms of tracking. In initial tests this method has shown reasonable agreement with VOF simulations for the wave maker problem and further validations will be presented.   

Phase-field simulations of free surface laminar flows

The laminar flow dynamics of free surfaces that includes wetting and rigid body motion is studied. The Navier-Stokes equations with added forces of surface tension and gravity governs the motion of fluid. Convective phase-field and composition equations that are derived based on Gibbs free energy functional governs the dynamics of interfaces. The coupled system is nondimensionalized and adaptive finite element method is implemented. Two problems are presented. First, a basic wetting of a liquid drop on a solid surface is simulated. The dependence of the apparent contact angle on the Capillary number is found to match known experimental data. The second problem involves the attraction/repulsion of two rigid particles due to capillary forces. The rate of attraction/repulsion of the rigid bodies influenced by different wetting conditions is analyzed.   

Numerical simulation of flow past transversely oscillating cylinder beneath a free surface

A computational study of laminar flow of a viscous incompresible fluid past transversely oscillating cylinder close to a free surface is performed. The integral form of unsteady two dimensional Navier-Stokes equations is only discretized in the fluid flow region using fixed Eulerian staggered grid. Well-posed boundary conditions are used at the inflow and outflow boundaries. The no-slip boundary conditions are prescribed at the solid boundary. At the free surface boundary conditions are described by neglecting the motion of ambient air. The volume of fluid method is used to track a moving free surface interface. A piecewise-linear interface reconstruction algorithm is used at each time step for determining the position of both the free surface and fluid-body interfaces. The reconstructed free surface is then advected using computed local velocity field based on a geometrical area-preserving volume of fluid advection algorithm. The numerical simulations are conducted at a fixed Reynolds number, $R=200$, and at displacement amplitude-to- cylinder diameter ratios of $A=0.25$ and $A=0.5$ when submergence depth-to-cylinder diameter ratio is $1.25$. Previously computed and observed flow fields around submerged cylinders are compared to current numerical results and good agreement is found.   

Numerical simulations of opto-hydrodynamics of fluid-fluid interfaces

Control of fluid-fluid interface deformation induced by the radiation pressure of a laser wave is of major concern for the development of numerous applications in microfluidics. Recent experimental work has shown that several deformation regimes can be identified and that, under certain conditions, hydrodynamic instability can occur leading to the formation of jets, drops and bridges. The purpose of the present work is to analyse the physics of deformation using direct numerical simulation of the hydrodynamics coupled to radiation. The configuration is that of a polarised and focused Gaussian laser beam impinging on an interface separating two immiscible viscous Newtonian liquids. A Boundary Integral Element Method was developed to solve the two-phase axisymmetric Stokes flow allowing a precise description of the interface shape and dynamics until final equilibrium is reached. Both linear and non-linear regimes are investigated and results are compared to experimental data as well as to the solution of the differential equation describing equilibrium. Strong non linear deformations are discussed.   

Flow meandering down an inclined plane: experiments and analytical modeling

We present an experimental and analytical study of a stream flowing down a flat, partially wetting inclined plane. Despite the practical importance of the problem, much remains to be discovered in the theory of meandering flows. We find that meandering flow regime, unlike braiding flows and rivulets, exists when there are fluctuations in the flow rate. From first principles, we derive a dynamic model of the system which is driven by noise. The model equations are solved exactly to predict the meandering exponent to be 1/6. It is surprising that this meandering exponent also corresponds well to 0.1-0.2 exponents experimentally observed in rivers.   

Circular Sheet Retraction

We present the results of a theoretical investigation of the dynamics of an axisymmetric sheet moving under the combined influence of curvature, viscous and inertial forces. The leading order equations are derived by exploiting the slenderness of the sheet. Particular attention is given to elucidating the importance of variations in sheet thickness, and to deriving the appropriate Trouton viscosity. Using this simplified model, we study the retraction of a circular, viscous film, extending the prior work by Brenner and Gueyffier.\footnote{Brenner {\&} Gueyffier, \textit{Phys. Fluids} \textbf{11}, 737-739, 1999.} We examine the dependence of the flow structure on the governing parameters, and we compare our numerical results with the experimental observations of Debregeas.\footnote{Debregeas, \textit{Phys. Rev. Lett.} \textbf{75}, 3886-3889, 1995.}   

Scale dependence of contact line computations

The shape and velocity of a sliding droplet are computed by solving the Navier--Stokes equation with free interface boundary conditions. The Galerkin finite element method is implemented in a 2D computation domain discretized using an unstructured mesh with triangular elements. The mesh is refined recursively at the corners (contact points). The stationary sliding velocity is found to be strongly dependent on grid refinement, which is a consequence of the contact line singularity resolved through the effective slip across the finite elements adjacent to the contact point. For small droplets, this dependence is well approximated by a theoretical estimates obtained using multiscale expansion and matching technique in lubrication approximation, where the corner element size is used as a microscale parameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where grids are invariably much more coarse than molecular scales on which the singularity is resolved. It is suggested that extrapolation to molecular scales should be used to obtain realistic results. In order to alleviate computational process, we carry out a separate computation of flow in the vicinity of the contact line which can be matched to macroscopic flow.