A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

A coupled fluid-elasticity model for the wave forcing of an ice-shelf

A mathematical model for predicting the vibrations of ice-shelves based on linear elasticity for the ice-shelf motion and potential flow for the fluid motion is developed. No simplifying assumptions such as the thinness of the ice-shelf or the shallowness of the fluid are made. The ice-shelf is modelled as a two-dimensional elastic body of an arbitrary geometry under plane-strain conditions. The model is solved using a coupled finite element method incorporating an integral equation boundary condition to represent the radiation of energy in the infinite fluid. The solution is validated by comparison with thin-beam theory and by checking energy conservation. Using the analyticity of the resulting linear system, we show that the finite element solution can be extended to the complex plane using interpolation of the linear system. This analytic extension shows that the system response is governed by a series of singularities in the complex plane. The method is illustrated through time-domain simulations as well as results in the frequency domain.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.     

Action of periodic surface pressure on an ice cover in the vicinity of a vertical wall

This paper presents the solution of the linear hydroelastic problem for steady forced vibrations of a semi-infinite ice cover under the effect of localized external load. The ice cover is simulated by a viscoelastic thin plate, the thickness of the fluid layer is assumed to be small, and the shallow water theory is used. The fluid is limited by a solid vertical wall, and the rectilinear edge of the elastic plate adjacent to the wall can be both free and clamped. The solution is obtained with the help of the Fourier integral transform. The behavior of the ice cover is studied depending on the frequency of the external load and boundary conditions on the edge of the plate. It is shown that, in the case of a free edge of the plate, there are considerable deflections on the edge, which could be comparable with deflections at the center of the pressure impact region. It is established that, due to the existence of wave movements of the type of edge waves, the external load energy is transferred to larger distances along the free edge, and there are significant bending moments on the edge of the clamped plate, which can lead to fracture of the ice cover with sufficiently great intensity of the external load.     

二维弹性结构入水冲击过程中的流固耦合效应

描述了一个研究弹性结构入水冲击过程中水弹性效应的数值方法,在弹性结构入水冲击过程中,流体域作用在结构上的水动力载荷由边界元法获得,而结构的弹性动力响应则由有限元方法求解,通过线性给离散Ｂｅｒｎｏｕｌｌｉ方程将有限元方程和边界元方程耦合到一起,从而获得了求解流场和结构动力响应的相互耦合的运动方程。在数值考虑了自由表面的非线性边界条件,通过引入射流单元以及最大射流厚度,较好地处理了冲击引起的射流问题。     

Numerical solutions for the flow of a fluid of grade three past an infinite porous plate

This paper presents a numerical study of the flow of an incompressible fluid of grade three past an infinite porous flat plate, subject to suction at the plate. This flow is governed by a non-linear differential equation that is particularly well suited to demonstrate the power and usefulness of different numerical techniques. In this work, the numerical solutions are obtained using a Runge-Kutta method of fourth order. The accuracy of the method for this problem is demonstrated.     

弹性地基上四边自由的各向异性矩形板

通过叠加法得到了弹性地基上的各向异性矩形板的一般解。每个叠加解被展成重傅立叶级数，其自身或其一阶导数在边界上的值被展成单傅立叶级数。利用控制微分方程和一些边界条件，每个叠加解被简化成用边界值的级数的系数表示的傅立叶级数。文后给出了弹性地基上的方板的挠曲面图。     

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.     

A low‐dimensional spectral approach for transient free‐surface flow inside thin cavities of symmetric shape

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside three‐dimensional cavities of general symmetric shape but of small thickness. The problem is closely related to the filling stage during the injection molding process. The moving domain is mapped onto a rectangular domain at each time step of the computation. A modified pressure is introduced, which in this case is governed by the Laplace's equation, and it is expanded in a Fourier series along the flow direction. The expansion coefficients are obtained using the finite‐difference method. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. Only a few modes are needed to secure convergence in general. It is found that the flow behaviour is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, cavity thickness, and the inlet flow. Copyright © 2002 John Wiley & Sons, Ltd.     

Plane problem of asymmetrical wave impact on an elastic plate

The problem of wave impact on the edge of an elastic horizontal plate is studied within the framework of the Wagner approach using the normal-modes method. The plate is governed by the Euler beam equation with simply supported ends. The liquid is assumed to be ideal and incompressible. The problem is coupled: the elastic and hydrodynamic characteristics of the impact process and the dimension of the contact region should be found simulatenously. An algorithm that permits a detailed study of the impact on an elastic plate is proposed. The phenomenon of unlimited increase of hydrodynamic loads owing to the plate flexibility (blockage) is revealed for fairly long plates. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirisk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 148–158, September–October, 1998.     

Bending of an Anisotropic Elliptic Plate on an Elastic Foundation

An approach is proposed to solve a stress–strain problem for anisotropic rigidly fixed plates on an elastic foundation. The problem is solved by the method of successive approximations. At each approximation, the deflection is represented as polynomials whose coefficients are determined from a system of linear algebraic equations. Study is made of the influence of the reinforcement angle and the modulus of subgrade reaction on the deflections and the bending moments in an orthotropic plate.     

A Lagrangian boundary element approach to transient three‐dimensional free surface flow in thin cavities

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The pressure, which in this case is governed by the Laplace's equation, is determined using the boundary element method. A fully Lagrangian approach is implemented for the tracking of the evolving free surface. The domain of computation is the projection of the physical domain onto the (x, y) plane. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. It is found that the flow behavior is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, and inlet flow pressure. Copyright © 2001 John Wiley & Sons, Ltd.     

非均布表面应力作用下薄板变形问题的求解

给出非均布表面应力作用下弹性薄板挠曲变形问题的控制方程及边界条件,通过与热应力问题进行物理比拟,对这一问题进行了求解,并采用这一方法对实验中观测到的局部弯曲现象进行了理论解释．     

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

薄板在冲击载荷下线弹性理想塑性响应的相似性研究

对于比例模型和原型使用不同弹塑性材料的冲击相似性问题,由于弹性和塑性阶段材料特性的差别及其在不同变形阶段的弹性塑性共存现象,将导致原有的结构冲击相似性理论失效。基于薄板冲击问题的理论模型,采用方程分析方法重新推导了材料线弹性以及理想刚塑性共存时的冲击响应的相似律。提出了一种能够同时考虑弹性变形和塑性变形的结构缩放响应相似性分析的厚度补偿方法,并利用数值分析验证了提出方法的适用性。分析结果表明,使用厚度补偿方法得到的比例模型结构响应能够准确地预测原型结构的冲击响应。     

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge–Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner et al. (2015, Journal of Fluid & Structures 52, 166–180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterized by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.     

Non-linear vibration analysis of an elastic plate subjected to heavy fluid loading in magnetic field

In the present study, the non-linear vibration of an elastic plate subjected to heavy fluid loading in an inclined magnetic field is investigated. The structural non-linearity, fluid non-linearity, and the effects of magnetic field are all incorporated in the formulations to derive the governing equation of the plate. The method of multiple scales is adopted to determine the eigenvalues and mode shapes of the linear vibration, and then the amplitude of the non-linear vibration response of the plate is calculated. Based on the assumptions of ordering and formulations of multiple scales, it can be concluded that the linear dynamic behavior of the plate under heavy fluid loading but weak near-resonant loading is influenced by the effects of the fluid loading, linear structural rigidity and linear magnetic field, furthermore, the non-linear dynamic behavior of the plate under heavy fluid loading but weak near-resonant loading is dominated and controlled by the effects of the fluid loading, non-linear structural rigidity and non-linear magnetic field. Both thick and thin plates are investigated; the contributions due to the structural non-linearity and acoustic linear radiation damping are of the same order for a rather thick plate. For a thin plate, the structural non-linearity completely controls the behavior of the plate, which implies that in this case the effect of fluid loading is considerably negligible. In general, it can be concluded that both the effects of magnetic field and structural non-linearity play important roles only on the first few modes of the plate.     

Particle removal from a surface by a bounded vortex flow

A bounded vortex flow consists of an axisymmetric vortex that is confined top and bottom between two plates (the “confinement plate” and “impingement plate”, respectively) and surrounded laterally by a swirling annular slot jet. The bottom of the vortex terminates on the boundary layer along the impingement plate and the top of the vortex is drawn into a suction port positioned at the center of the confinement plate. The circumferential flow within the annular jet is important for supplying circulation to the central wall-normal vortex. This flow field is proposed as a method for mitigation of dust build-up on a surface, where the vortex–jet combination supplements the more traditional vacuum port by enhancing the surface shear stress and related particle transport rate. The paper reports on a computational study of the velocity field and particle transport by a bounded vortex flow. Fluid flow computations are performed using a finite-volume approach for an incompressible fluid and particle transport is simulated using a discrete-element method. Computations are performed for different values of two dimensionless parameters – the ratio of the plate separation distance and the average radial location of the jet inlet (the dimensionless confinement height) and the ratio of flow rate withdrawn at the suction outlet and that injected by the jet (the flow rate ratio). For small values of the flow rate ratio, the impinging jet streamlines pass down to the boundary layer along the bottom surface and then travel up the vortex core. By contrast, for large values of flow rate ratio, the annular jet is quickly entrained into the suction outlet and no wall-normal vortex is formed. Particles are observed to roll along the impingement surface in a direction determined by the fluid shear stress lines. Particles roll outward when they lie beyond a separatrix curve of the surface shear stress lines, where particles within this separatrix curve roll inward, piling up at the center of the flow field. A toroidal vortex ring forms for the small confinement height case with flow rate ratio equal to unity, which yields double separatrix curves in the shear stress lines. The inward rolling particles intermittently lift up due to collision forces and burst away from the impingement surface, eventually to become entrained into the flow out the suction port or resettling back onto the impingement surface.