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

Advances in direct numerical simulation for MHD modeling of free surface flows

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

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.

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.

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.

Multiphysics numerical modelling of backward flow forming process of AISI 5140 steel

Computational fluid dynamics (CFD) is increasingly becoming the tool of choice for researchers and engineers facing complex fluid flow problems. One area of CFD that has attracted significant attention in recent years is the development of meshless or mesh-free methods[1]. These methods have so far been used in many applications like fracture mechanics, astrophysics and fluid flow. The key advantage of meshless methods over mesh-based methods like finite volume (FV) and finite element (FE) methods is its non-requirement of pre-specified connectivity between nodes in order to derive approximation and interpolation. This leads to the ability of finite difference (FD) methods to treat irregular domains and boundaries. A direct result of this is another advantage of mesh-free methods having the adaptability for problems involving large movement or wholesale motion of boundaries or embedded bodies. In this area, it has the edge over composite or overset grid methods in being able to use only one coordinate frame.A convecting meshless scheme for boundary driven fluid flow is described. This scheme can be applied to self-propulsion problems, or, in general, flows involving motion of rigid or deformable bodies embedded within the flow.The generalized finite difference method[2] (GFD) was chosen this scheme. A meshless cloud of nodes is embedded around the body surface along with the structured Cartesian background nodes for the entire flow field. By incorporating a subdomain for each node and its neighbours, the use of Taylor series expansion enables the derivatives to be found. Two forms of incompressible momentum equations are used: the standard Eulerian form in primitive variables for the structured nodes and the Arbitrary Lagrangian-Eulerian (ALE) form for the meshless cloud of nodes. GFD with moving least squares approximation is used for spatial discretization of the convecting meshless nodes. The projection method with Crank-Nicolson discretization is used to achieve second order accurate time discretization. Following prescribed body motion or fluid structure interaction, the meshless cloud acquire new locations in the next time step and the momentum equations in ALE form can then be solved.Test cases were performed to verify code usability and method accuracy and convergence. The simulation of the decaying vortex has been a popular benchmark for the gauging of accuracy. Basically the exact solutions of the time-dependent function values are known and both residues and absolute errors were calculated. Solving using different grid sizes proved the present method to be of second order spatial accuracy. The meshless method was then tried on the driven cavity flow at Reynolds number 1000. The results were well-compared with popular benchmark results[4]. In considering the effects of implementing the convecting meshless nodes across a normal flow field, the decaying vortex was again simulated, with a moving patch of nodes within the Cartesian field. The residues were well controlled and the errors were not significantly increased. Finally, a couple of external flows cases were simulated, basically on a flow past a stationary bluff body and a flapping ellipse inside an enclosed body of fluid.Future works in the pipeline include writing a parallel code for larger scale simulation and simulating oscillating cylinders and flapping of elliptic aerofoil(s).

Transient solution of 3D free surface flows using large time steps

This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of large deformation quasi-static and dynamic problems. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. Time integration relations for the dynamic equations are also derived. Several quasi-static and dynamic large deformation applications are solved and presented.

유체자유수면의 동적거동을 합리적으로 예측하기 위해서는 비선형 특성을 보이는 자유수면의 동역학적 경계조건을 고려해야할 뿐만 아니라 시간에 따라 변화하는 자유수면의 위치변화에 따른 운동학적 경계조건을 고려하여야 한다. 이러한 문제는 대상구조물이 3차원이 될 경우 더욱 복잡해지므로 3차원 비선형 유체자유수면의 해석은 이론해의 도출이 어려우며 수치해석 방법을 이용하는 것이 효과적이다. 본 연구에서는 수치해석 안정성이 높고 3차원 문제에서도 하나의 변수로 유체거동을 모사할 수 있는 arbitrary Lagrangian-Eulerian approach 를 경계요소에 적용하여 효율적이며 안정적인 유체 대변형 해석기법을 개발하였다. 개발된 기법은 향후 자유수면의 비선형 효과를 고려한 유체-구조물 상호작용 해석에 효과적으로 적용할 수 있을 것으로 판단된다. The solution to a liquid sloshing problem is challenge to the field of engineering. This is not only because the dynamic boundary condition at the free surface is nonlinear, but also because the position of the free surface varies with time in a manner not known a priori. Therefore, this nonlinear phenomenon, which is characterized by the oscillation of the unrestrained free surface of the fluid, is a difficult mathematical problem to solve numerically and analytically. In this study, three-dimensional boundary element method(BEM), which is based on the so-called an arbitrary Lagrangian-Eulerian(ALE) approach for the fluid flow problems with a free surface, was formulated to solve the behavior of the nonlinear free surface motion. An ALE-BEM has the advantage to track the free surface along any prescribed paths by using only one displacement variable, even for a three-dimensional problem. Also, some numerical examples were presented to demonstrate the validity and the applicability of the developed procedure.

The flow of a liquid film over an inclined plane is frequently found in nature and industry, and, under some conditions, instabilities in the free surface may appear. These instabilities are initially two-dimensional surface waves, known as Kapitza waves. Surface waves are important to many industrial applications. For example, liquid films with surface waves are employed to remove heat from solid surfaces. The initial phase of the instability is governed by the Orr-Sommerfeld equation and the appropriate boundary conditions; therefore, the fast and accurate solution of this equation is useful for industry. This paper presents a spectral method to solve the Orr-Sommerfeld equation with free surface boundary conditions. Our numerical approach is based on a Galerkin method with Chebyshev polynomials of the first kind, making it possible to express the Orr-Sommerfeld equation and their boundary conditions as a generalized eigenvalue problem. The main advantages of the present spectral method when compared to others such as, for instance, spectral collocation, are its stability and its readiness in including the boundary conditions in the discretized equations. We compare our numerical results with analytical solutions based on Perturbation Methods, which are valid only for long wave instabilities, and show that the results agree in the region of validity of the long-wave hypothesis. Far from this region, our results are still valid. In addition, we compare our results with published experimental results and the agreement is very good. The method is stable, fast, and capable to solve initial instabilities in free surface flows.

We are interested in the numerical solution of a two-dimensional fluid-structure interaction problem.A special attention is paid to the choice of physically relevant inlet boundary conditions for the case of channel closing.Three types of the inlet boundary conditions are considered.Beside the classical Dirichlet and the do-nothing boundary conditions also a generalized boundary condition motivated by the penalization prescription of the Dirichlet boundary condition is applied.The fluid flow is described by the incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian (ALE) form and the elastic body creating a part of the channel wall is modelled with the aid of linear elasticity.Both models are coupled with the boundary conditions prescribed at the common interface.The elastic and the fluid flow problems are approximated by the finite element method.The detailed derivation of the weak formulation including the boundary conditions is presented.The pseudo-elastic approach for construction of the ALE mapping is used.Results of numerical simulations for three considered inlet boundary conditions are compared.The flutter velocity is determined for a specific model problem and it is shown that the boundary condition with the penalization approach is suitable for the case of the fluid flow in a channel with vibrating walls.

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.