Analytical modelling for a three-dimensional hydrofoil with winglets operating beneath a free surface

The current study focuses on establishing a theoretical lifting surface model for predicting the hydrodynamic loads acting on the three-dimensional hydrofoil with winglets, which is considerably influenced by the proximity to the free surface through finding the three-dimensional Green’s function for the planar and vertical horseshoe vortices operating below a free surface. The hydrofoil surface is decomposed into a finite number of elements along the span direction and the chord directions, each of which can then be represented by a horseshoe vortex. The linearized free surface boundary condition is applied to analyze the influence of the free surface on the hydrofoil as well as the winglets. The thickness problem is considered using the source distribution among the hydrofoil and winglets surfaces and the analytical Green’s function that satisfies the linearized free surface boundary condition is used. As a sample application, numerical examples were conducted to show the performance of the hydrodynamic characteristics for the hydrofoil with winglets as a function of the Froude number. It was concluded that there are significant efficiency benefits from using winglets inside the free surface proximity effect. These results are substantiated by the comparison with the available published data.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Permanent Wave Structures on an Open Two Layer Fluid

There is a significant nonlinear interaction between fast free surface waves and slow interface waves when the group velocity of the free surface waves is the same as the phase velocity of the interface waves. This interaction leads to permanent wave structures consisting of a wave group of permanent envelope on the free surface and a wave of permanent shape on the interface. A theory is developed for periodic permanent wave structures of this type, from which solutions are found numerically. The theory includes all significant quadratic nonlinear interactions between free surface harmonics and interface harmonics, as well as between the interface harmonics themselves. It is found that there is a sequence of forms of differing free surface group structure.     

Exact Free Surface Flows for Shallow Water Equations II: The Compressible Case

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi-series. Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when γ = 1, we again find simple wave and time-dependent solutions.     

Modelling free surfaces in oscillating pipe flows

An oscillating pipe flow with a free surface is investigated numerically and experimentally. The pipe diameter is 12mm. Due to this small diameter capillary forces play an important role. Therefore special attention has to be paid to the flow field near the free surface. The numerical model is based on the fundamental flow equations. The free surface is resolved according to the volume-of-fluid method. The model equations are solved on a moving grid. In the experiment, pictures of the flow field are taken near the free surface. The effects occuring near the interface will be presented here. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical procedure for viscous free surface flows

We present a numerical procedure for two-dimensional unsteady viscous free surface flow problems with surface tension. The procedure is based on a finite difference approach to a primitive variable formulation; a coordinate transformation is used to transform the irregularly shaped flow domain onto a fixed rectangular domain. The procedure is tested on a standing wave problem and a cavity flow problem with a free surface. Satisfactory numerical solutions are obtained for both problems for Reynolds numbers up to 200.     

Inequalities for single crystal rod growth by edge-defined film-fed growth (E.F.G.) technique

A boundary value problem in the case of the second order axi-symmetric Young-Laplace differential equation (some of whose solutions describe the static meniscus free surface, i.e. the static liquid bridge free surface between the shaper and the crystal, occurring in single crystal rod growth) is analyzed. The analysis concerns the dependence of the solution of an initial value problem of the equation on a parameter p (the controllable part of the pressure difference Δp across the free surface). Inequalities are established for p which are necessary or sufficient conditions for the existence of a solution which represents a stable and convex free surface of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in the NdYAG laser single crystal rod growth from the melt by edge-defined film-fed growth (E.F.G.) technique. This kind of inequalities can be useful in the experiment planning and technology design.     

Rotational waves generated by current‐topography interaction

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.     

Nonlinear free surface condition due to second order diffraction by a pair of cylinders

An analysis of the non-homogeneous term involved in the free surface condition for second order wave diffraction on a pair of cylinders is presented. In the computations of the nonlinear loads on offshore structures the most challenging task is the computation of the free surface integral. The main contribution to this integrand is due to the nonhomogeneous term present in the free surface condition for second order scattered potential. In this paper, the free surface condition for the second order scattered potential is derived. Under the assumption of large spacing between the two cylinders, waves scattered by one cylinder may be replaced in the vicinity of the other cylinder by equivalent plane waves together with non-planner correction terms. Then solving a complex matrix equation, the first order scattered potential is derived and since the free surface term for second order scattered potential can be expressed in terms of the first order potentials, the free surface term can be obtained using the knowledge of first order potentials only.     

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.     

A mapping technique for numerical computations of bed evolutions

Free surface flow is one of the most difficult problems in engineering to be solved, since velocity and pressure fields depend on the free surface. On the other hand, the position of the free surface is unknown previously. Furthermore, the boundary condition on the free surface is expressed by a complicated equation. In an alluvial stream, where the boundaries of the domain are not fixed, addition of free surface at the bed will increase this difficulty. A domain mapping technique is developed in this paper to study the bed evolutions. The flow is considered 2D, choosing two coordinates in streamwise and upward directions. With a proper transformation, the hydrodynamics and sediment transport governing equations in irregular domain will be mapped into a simple rectangular one. The new domain can be discretize by finite elements. The transformed governing equations are solved to obtain desired variables in the mapped domain. With a proper transformation, there is no need of inverse mapping to obtain the free water surface profile and bedform evolution and migration in the actual domain. The model has been applied to streams with movable bed and the results show a good agreement with the experimental experiences.     

Fluid flow through a vertical slot under gravity

The flow of fluid from a horizontal uniform channel of finite depth into a vertical slot under the influence of gravity is considered. The shapes of a top and bottom free surface are computed for a range of parameter values. Solutions in which the top free surface attaches smoothly to the vertical wall were only found for Froude numbers greater than or equal to one, though solutions were found for all gap sizes.     

Finite element simulation for material accumulation and welding processes including a free melt surface

A modeling and simulation approach for problems with solid-liquid-solid phase transitions and a free surface, feasible for material accumulation processes based on laser-based free form heading and welding processes for joining different metallic materials is presented. Both named processes are modeled within the framework of continuum mechanics by coupling the Stefan problem with the Navier-Stokes equations including a free capillary surface. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stationary viscous flows at zero surface tension

Stationary solutions of free boundary problems for the Navier-Stokes equations are considered with and without surface tension. The linearized problem in a half-space is studied. Exact solutions of the Poiseuille type are obtained. Properties of the free boundary smoothness in 3-D and 2-D cases are discussed.     

3D SPH flow predictions and validation for high pressure die casting of automotive components

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three-dimensional fluid flow with significant free surface fragmentation and splashing. A Lagrangian simulation technique that is particularly well suited to modelling HPDC is smoothed particle hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling the accurate prediction of fluid flows involving complex free surface motion.     

An equivalent pipe network model for free surface flow in porous media

Free surface flow analysis in porous media is challenging in many practical applications with strong non-linearity. An equivalent pipe network model is proposed for the simulation and evaluation of free surface flow in porous media. On the basis of representative elementary volume with homogeneous pore-scale patterns, the pore space of the homogeneous isotropic porous media is conceptualized as a collection of capillary tubes. According to Hagen-Poiseulle's law and flux equivalence principle, equivalent hydraulic parameters and unified governing formulations for the pipe network model are deduced. The two-dimensional free surface flow problem is reduced to a one-dimensional problem of pipe networks and a one-dimensional procedure based on the finite element method is then developed by introducing a continuous penalized Heaviside function. The proposed equivalent pipe network model is verified with results from numerical solutions and laboratory-measured data available in the literature, and good agreements are obtained. The proposed equivalent pipe network model is shown to be effective in analyzing the free surface flow in porous media. The numerical results also indicate that the proposed equivalent pipe network model has weak sensitivity of the mesh size and penalty parameters.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

The Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface tension is either taken into account at both free boundaries or neglected at both. We are concerned with the Rayleigh–Taylor instability, so we assume that the upper fluid is heavier than the lower fluid. When the surface tension at the free internal interface is below a critical value, which we identify, we establish that the problem under consideration is nonlinearly unstable.     

水箱突然起动的水动压力问题

在本文中,应用Lagrange方法讨论水箱突然起动的水动压力问题.求得了自由面形状和作用在水箱壁上的动态压力系数.也证实了在自由面附近,压力系数的奇性仅是对数奇性.     

A novel algorithm based on parameterization method for calculation of curvature of the free surface flows

In this paper, a new approach based on parameterization method is presented for calculation of curvature on the free surface flows. In some phenomena such as droplet and bubble, surface tension is prominent. Therefore in these cases, accurate estimation of the curvature is vital. Volume of fluid (VOF) is a surface capturing method for free surface modeling. In this method, free surface curvature is calculated based on gradient of scalar transport parameter which is regarded as original method in this paper. However, calculation of curvature for a circle and other known geometries based on this method is not accurate. For instance, in practice curvature of a circle in interface cells is constant, while this method predicts different curvatures for it. In this research a novel algorithm based on parameterization method for improvement of the curvature calculation is presented. To show the application of parameterization method, two methods are employed. In the first approach denoted by, three line method, a curve is fitted to the free surface so that the distance between curve and linear interface approximation is minimized. In the second approach namely four point method, a curve is fitted to intersect points with grid lines for central and two neighboring cells. These approaches are treated as calculus of variation problems. Then, using the parameterization method, these cases are converted into the sequences of time-varying nonlinear programming problems. With some treatments a conventional equivalent model is obtained. It is finally proved that the solution of these sequences in the models tends to the solution of the calculus of variation problems. For verification of the presented methods, curvature of some geometrical shapes such as circle, elliptic and sinusoidal profile is calculated and compared with original method used in VOF process and analytical solutions. Finally, as a more practical problem, spurious currents are studied. The results showed that more accurate curve prediction is obtained by these approaches than the original method in VOF approach.