Chapter 6 - Free Surface and Buoyancy Driven Flows

Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation

Since free surface flows are seen in nature phenomena and artificial systems, the analysis for these flows has been more important in mechanical engineering. However, theoretical analyses are almost impossible because the boundary conditions are nonlinear equations in analyzing the free surface flows. Thus, numerical analyses are needed. Finite element method (FEM) which can be used for unstructured grid can easily treat boundary conditions. Thus, FEM is used in many technological fields. In this thesis, free surface flow problems were analyzed using the finite element method. In solving these problems, an arbitrary Lagrangian-Eulerian (ALE) kinematical description of the fluid domain was adopted, in which the nodal point can be displaced independently of the fluid motion. The ALE method was introduced into the Generalized Simplified Marker and Cell (GSMAC) method. Damping out the alternating errors was examined in this scheme.

A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

A numerical model based on the two-phase flow model for incompressible viscous fluid with a complex free surface has been developed in this study. The two-step projection method is employed to solve the Navier–Stokes equations in the numerical solutions, and finite difference method on a staggered grid is used throughout the computation. The two-order accurate volume of fluid (VOF) method is used to track the distorted and broken free surfaces. The two-phase model is first validated by simulating the dam break over a dry bed, in which the numerical results and experimental data agree well. Then 2-D fluid sloshing in a horizontally excited rectangular tank at different excitation frequencies is simulated using this two-phase model. The results of this study show that the two-phase flow model with VOF method is a potential tool for the simulation of nonlinear fluid sloshing. These studies demonstrate the capability of the two-phase model to simulate free surface flow problems with considering air movement effects.

Comparison of wall boundary conditions for numerical viscous free surface flow simulation

A diffuse interface method for complex three-dimensional free surface flows

Nonlinear solutions of free surface flow under a sluice gate are studied. Upstream, the fluid is assumed to be infinitely deep, and the gate is at an angle to the horizontal axis. The flow emerges from the gate and produces a uniform flow far downstream. The problem is solved numerically by a boundary element method derived from an integral equation along the free surface. An analytical function is constructed, related to the upstream flow, so that the integral equation is solvable. As a result, a free surface flow with smooth detachment from the edge of the gate is obtained for relatively large upstream Froude numbers. As the Froude number decreases, a free surface with back flow near the edge of the gate occurs, until at a certain Froude number a stagnation point occurs. References D. D. Frangmeier, T. S. Strelkoff, Solution for gravity flow under a sluice gate, ASCE J. Engng. Mech. Div. , 94 , 1968, 153--176. B. E. Lorock, Gravity-affected flow from planer sluice gate, ASCE J. Engng. Mech. Div. , 96 , 1969, 1211--1226. Y. K. Chung, Solution of flow under sluice gates, ASCE J. Engng. Mech. Div. , 98 , 1972, 121--140. J. Asavanant, J.-M. Vanden-Broeck, Nonlinear free surface flows emerging from vessels and flows under a sluice gate, J. Austral. Math. Soc. , 38 , 1996, 63--86. doi:10.1017/S0334270000000473 J.-M. Vanden-Broeck, Numerical calculations of the free-surface flow under a sluice gate, J. Fluid Mech. , 330 , 1996, 339--347. http://journals.cambridge.org/abstract_S0022112096003849 B. J. Binder, J.-M. Vanden-Broeck, Free surface flows past surfboards and sluice gates, Eur. J. Appl. Maths. , 16 , 2005, 601--619 doi:10.1017/S0956792505006443 B. J. Binder, J.-M. Vanden-Broeck, The effect of disturbances on the flows under a sluice gate and past an inclined plate, J. Fluid Mech. , 576 , 2007, 475--490 doi:10.1017/S0022112007004806 J.-M. Vanden-Broeck, E. O. Tuck, Flow near the interaction of a free surface with a vertical wall, SIAM J. Appl. Math. , 54 , 1994, 1--33. doi:10.1137/S0036139992240838 L. H. Wiryanto, J. Widjaja, B. Supriyanto, Free-surface flow under a sluice gate from deep water, Bul. Malay. Math. Soc. , 34 , 2011, 601--609. http://www.emis.de/journals/BMMSS/vol34_3_17.html L. H. Wiryanto, E. O. Tuck, A back-turning jet formed by a uniform shallow stream hitting a vertical wall, Proc. Int. Conf. on Dif. Eqs. , Kluwer Academic Press, 1997. L. H. Wiryanto, E. O. Tuck, An open-channel flow meeting a barrier and forming one or two jets, J. Austral. Math. Soc. , 41 , 2000, 458--472. doi:10.1017/S0334270000011760 L. H. Wiryanto, Small Froude number solutions of flow caused by a line source, J. Indones. Math. Soc. , 6 , 2000, 1--6. G. C. Hocking, L. K. Forbes, A note on the flow induced by a line sink beneath a free surface, J. Austral. Math. Soc. B , 32 , 1991, 251--260. doi:10.1017/S0956792507006924 G. C. Hocking, L. K. Forbes, Subcritical free-surface flow caused by a line source in a fluid of finite depth, J. Eng. Math. , 26 , 1992, 455--466. doi:10.1007/BF00042763 L. H. Wiryanto, Zero gravity of a jet emerging from a slit, J. Indones. Math. Soc. , 12 , 2006, 89--98. L. M. Milne-Thomson, Theoretical hydrodynamics . Dover Publications, Inc., 1968, 274--281.

Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity

Liquid sloshing is a common phenomenon in the transporting of liquid tanks. Liquid waves lead to fluctuating forces on the tank wall. If these fluctuations are not predicted or controlled, they can lead to large forces and momentum. Baffles can control liquid sloshing fluctuations. One numerical method, widely used to model the liquid sloshing phenomena is Smoothed Particle Hydrodynamics (SPH). Because of its Lagrangian nature, SPH is suitable for simulating free surface flow. In the present study, a relatively accurate Incompressible SPH (ISPH) method improved by kernel gradient correction tensors, particle shifting algorithms, turbulence viscosity calculations, and free surface particle detectors is applied for the free surface flow modeling. In comparison to the other SPH Simulations and experimental data, these results show that the present algorithm is effective for simulating free surface problems. The present algorithm has been applied to simulate liquid sloshing phenomena, while the aim of this study is the investigation of vertical and horizontal baffle effects on the control and damping of liquid sloshing. Results show that for vertical baffles, baffle size has a major role in sloshing fluctuation damping. For horizontal baffles, also including size, the baffle base position has a significant role in liquid sloshing fluctuation damping. When horizontal baffle is near the free surface, sloshing fluctuation-damping increases.

A fixed-grid model for simulation of a moving body in free surface flows

We present a residual-based turbulence model for problems with free surfaces. The method is derived based on variational multiscale ideas that assume a decomposition of the solution fields into overlapping scales that are termed as coarse and fine scales. The fine scales are further split hierarchically into fine-scales level-I and fine-scales level-II. The hierarchical variational problems that govern the two fine-scale components are modeled employing bubble functions approach. The model for level-II scales is variationally embedded in the mixed field level-I problem to yield a stable level-I formulation. Subsequently, the model for level-I scales that in fact constitutes the fine-scale turbulence model is then variationally injected in the coarse-scale variational form. A significant feature of the method is that it does not contain any embedded tunable parameters. To accommodate the moving boundaries we cast the formulation in an arbitrary Lagrangian–Eulerian frame of reference. The free surface boundary condition is imposed weakly which results in a formulation that conserves the volume of the fluid. A variety of benchmark problems show the accuracy and range of applicability of the proposed formulation and results are compared with published data. A wavy bed problem is investigated to show the interaction of turbulence generated at the bottom surface with the free surface thereby leading to irregular free surface elevations.

In this paper we establish a link between the sigma transformation approach and the arbitrary Lagrangian–Eulerian (ALE) approach. For that purpose we introduce the ALE‐sigma (ALES) approach, which consists in an ALE interpretation of the sigma transformation. Taking advantage of this new approach, we propose a general ALES transformation, allowing for a great adaptability of the vertical discretization and therefore overcoming some drawbacks of the classical sigma transformation. Numerical results are presented, showing the advantages of this general coordinate system, as, for example, a better representation of horizontal stratifications. Copyright © 2008 John Wiley & Sons, Ltd.

Towards non-intrusive reduced order 3D free surface flow modelling

A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation

In this paper we will present an algorithm to perform free surface flow simulations with the lattice Boltzmann method on adaptive grids. This reduces the required computational time by more than a factor of three for simulations with large volumes of fluid. To achieve this, the simulation of large fluid regions is performed with coarser grid resolutions. We have developed a set of rules to dynamically adapt the coarse regions to the movement of the free surface, while ensuring the consistency of all grids. Furthermore, the free surface treatment is combined with a Smagorinsky turbulence model and a technique for adaptive time steps to ensure stable simulations. The method is validated by comparing the position of the free surface with an uncoarsened simulation. It yields speedup factors of up to 3.85 for a simulation with a resolution of 4803 cells and three coarser grid levels, and thus enables efficient and stable simulations of free surface flows, e.g. for highly detailed physically based animations of fluids.