Numerical investigation of the interaction of a vortex dipole with a deformable plate

Energy harvesting from coherent fluid structures is a current research topic due to its application in the design of small self-powered sensors for underwater applications. The impact of a vortex dipole with a deformable cantilevered plate at the plate tip is herein studied numerically using a strongly coupled staggered fluid–structure interaction algorithm. Three dipole Reynolds numbers, Re=500, 1500, and 3000, are investigated for constant plate properties. As the dipole approaches the plate, positive vorticity is produced on the impact face, while negative vorticity is generated at the tip of the plate. Upon impact, the dipole splits into two, and two secondary dipoles are formed. The circulation and, therefore, the trajectories of these dipoles depend on both the Reynolds number and the elasticity of the plate, and these secondary dipoles may return for subsequent impacts. While the maximum deflection of the plate does not depend significantly on Reynolds number, the plate response due to subsequent impacts of secondary dipoles does vary with Reynolds number. These results elucidate the strong interdependency between plate deformation and vortex dynamics, as well as the effect of Reynolds number on both.     

Nonstationary motion of a shell on the surface of a heavy fluid

A three-dimensional nonstationary problem of vibrations of a flexible shell moving on the surface of an ideal heavy fluid. The forces due to surface tension are ignored. The problem is formulated in the space of the acceleration potential. The potential of the pulsating source is found by solving the Euler equation and the continuity equation taking into account the free-surface conditions (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions on the surface of the shell. Formulas for determining the shape of gravity waves on the fluid surface and the natural frequencies of vibrations of the shell are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 66–75, July–August, 2009.     

Separation Impact on a Plate Floating on the Surface of an Ideal Incompressible Fluid in a Bounded Tank

The paper studies the planar problem of separation impact on a plate floating on the surface of an ideal incompressible fluid in a bounded tank. The problem is solved using an asymptotic method under the assumption that the immovable rigid walls of the tank are at a large distance from the plate. It is concluded that the tank walls of arbitrary shape have an ambiguous effect on the fluid particle separation zone formed on the plate surface is revealed. Examples of solutions are given.     

On the radiation pattern of a paraboloid of revolution

Summary The radiation pattern of a paraboloid of revolution is determined by using the mathematical form of Huygens' principle where the field intensities in a given domain are expressed in their tangential components on the boundary. These tangential components are supposed to have the values which follow from the reflection of a plane wave at an infinite metal plate, coinciding with the tangent plane to the surface of the paraboloid. Expressions involving infinite series of Bessel functions are obtained which give the intensities of the distant field when an electric dipole is placed at the focus. Approximations are made for the main lobe in the planes perpendicular to and through the dipole. Finally Zuhrt's method leads to the same results as the approximations given in the present paper.     

Behavior of a Semi-Infinite Ice Cover Under a Uniformly Moving Load

This paper consideres the behavior of a semi-infinite ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a load moving with constant velocity along the edge of the cover at some distance from it. The ice cover is modeled by a thin elastic plate of constant thickness. In a moving coordinate system, the deflection of the plate is assumed to be steady. An analytic solution of the problem is obtained using the Wiener–Hopf technique. The wave forces, the deflection of the plate, and the elevation of the free surface of the fluid at different velocities of the load are investigated.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Influence of an elastic plate on the motion of an inhomogenfous fluid

The influence of a thin elastic isotropic plate on the wave motion of an inhomogeneous fluid originating under the effect of external periodic perturbations is investigated. The fluid density increases constantly with depth. Analogous problems have been examined for an inhomogeneous fluid without a plate in [1, 2] and with a plate on the surface of a homogeneous fluid in [3–5].Sevastopol'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–67, January–February, 1972.     

STUDY ON THE DYNAMIC BEHAVIOR OF AXIALLY MOVING RECTANGULAR PLATES PARTIALLY SUBMERSED IN FLUID

The natural frequencies, complex modes and critical speeds of an axially moving rectangular plate, which is partially immersed in a fluid and subjected to a pretension, are investigated. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The subsection functions are used to describe the discontinuous characteristics of the system due to partial immersion. The classical thin plate theory is adopted to formulate the equations of motion of a vibrating plate. The velocity potential and Bernoulli's equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be equivalent to the added mass on the plate. The effects of distance ratio, moving speed, immersed-depth ratio, boundary conditions, stiffness ratio and aspect ratio of the plate as well as the fluid-plate density ratios on the free vibrations of the moving plate-fluid system are investigated.     

Drag of a Plate Planing on the Shallow Water with Formation of Waves

The plane problem of the plate planing at a constant velocity on the surface of a heavy, ideal, incompressible, finite-depth fluid is considered. The approximate, depth-independent expression for the force acting on the plate is derived from the linear distribution of the fluid velocity along the plate and the height of the flow stagnation point, without regard for jet formation near the leading edge. In this approximate formulation the plate drag depends on its velocity and the trailing edge immersion and does not depend on the planing angle. Experiments and numerical calculations in the exact formulation are performed in the near-critical flow regimes. It is shown that the wave patterns in the experiments and numerical calculations coincide, the formula for the drag being in agreement with the numerical experiments. An approximate criterion of the formation of waves going away from the plate in the forward direction is proposed.     

The effect of end conditions on the vortex-induced vibration of cylinders

In the present investigation we study the effect of end conditions on the vortex-induced vibration of an elastically mounted rigid cylinder. This work was triggered by some initial controlled vibration experiments which showed that spanwise end conditions can have a large effect on measured fluid forces on a cylinder, and this suggested that some of the disparity amongst previous free vibration studies may possibly be attributed to differences in end conditions. In the principal experiments here, we are concerned with a vertical cylinder piercing the clean free surface of a water channel, and attached to a carriage system mounted atop the channel. The upper end of the submerged cylinder is thus the free surface, while the lower end is manipulated to yield three different conditions, namely: an attached endplate; an unattached endplate fixed to the channel floor (with a variable gap between cylinder and plate); and a condition of no endplate at all. Interestingly, we find that the free vibration response for the attached and unattached endplate cases were nearly identical. One expectation was that the case without an endplate would lead to a flow around the end of the body, modifying the vortex dynamics, and thereby reducing the correlation of the induced fluid forces on the body. Surprisingly, over the entire response plot, the vibration amplitude is markedly higher in the absence of an endplate, with the exception of the peak amplitude, which remains nearly unchanged. Unexpectedly, the vibrations become much more steady at flow velocities in the vicinity of the peak response, if the endplate is removed. In a further set of experiments, we undertake controlled vibration, where we vary the gap between cylinder and endplate. We discover a large discontinuous jump in the magnitude of fluid excitation, when the gap exceeds 15% of a diameter. For larger gaps, the fluid excitation becomes independent of the gap size, effectively equivalent to having no plate at all. This study is consistent with some of the disparity between the character of vibration response plots in previous studies, if one takes into account the particular end conditions chosen in those studies.     

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

在水槽中的加速平板问题

本文利用Lagrange方法分析在水槽中平板水平地向静止流体运动的问题,非线性自由面条件能精确满足,所得的结果大致与[2]符合。     

Radiation effects on the thermal boundary layer flow over a moving plate with convective boundary condition

The steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition and in the presence of thermal radiation is investigated in this paper. Under certain conditions, the present problem reduces to the classical Blasius and Sakiadis problems. The effects of radiation and convective parameters on the thermal field are thoroughly examined and discussed. Dual solutions are found to exist when the plate and the fluid move in the opposite directions.     

Two-dimensional problem of periodic loading of an elastic plate floating on the surface of an infinitely deep fluid

The action of external periodic pressure on an elastic plate floating on the surface of a fluid assumed to be ideal and incompressible is examined by the method of normal modes in the linear formulation. The behavior of the matrix of coefficients of the hydrodynamic load on the plate is considered in detail for different frequencies. The behavior of the plate under localized periodic loading is compared for the cases of a heavy fluid with a finite or infinite depth and for a weightless infinite-depth fluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 61–72, May–June, 2005.     

Methods of Solving Nonlinear Problems of Hydrodynamic Impact in Bounded Regions

The plane problem of separation impact on a plate floating on the surface of an ideal incompressible fluid in a bounded vessel is considered. In this problem the zone of contact between the body and the fluid is not known in advance and must be determined together with the fluid flow. As a result, the problem formulated is nonlinear and belongs to the class of free-boundary problems. The effect of vessel walls of different shapes on the fluid particle separation zone formed on the plate surface is studied. As examples, the problems for a layer or a truncated circular meniscus are considered.     

Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid

A plane unsteady-state linear problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. The deflection of the plate and the velocity of its points are known at the initial moment of time. The fluid occupies the lower halfplane, and its boundary outside the plate is free. The plate which is the bottom of a structure immersed in the fluid with a constant velocity is modeled by an Euler beam. At the initial stage of immersion, when the displacement of the structure is much smaller than the length of the plate, the plate deflection and the distribution of bending stresses in it are determined. The model used allows one to estimate the maximum stresses occurring in the elastic plate during its impact on water and to predict the moment and site of its occurrence. Calculations are performed under the conditions of the experiment carried out in MARINTEX (Norway). Qualitative agreement between the numerical and experimental results is shown. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 150–160, May–June, 1999.     

Simulation of viscous fluid flow with consideration of boundary outflow and pressure drop

The problem of modeling a viscous fluid flow over the surface of a plate is considered when the pressure changes along the longitudinal coordinate according to a linear law. The corresponding boundary conditions are formulated for this problem. The Navier-Stokes equations are solved exactly in the problem of flow past the plate for the case of fluid outflow and a longitudinal pressure drop. Several formulas to determine the velocity profile are derived. The limiting cases are analyzed to study the consistency of various models. The corresponding pressure conditions are proposed for the case when the Navier-Stokes system has a known exact solution.     

COUPLED VIBRATORY CHARACTERISTICS OF A RECTANGULAR CONTAINER BOTTOM PLATE

The natural frequencies of an elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface are studied. The fluid is assumed to be incompressible, inviscid and irrotational, and the effect of surface waves is neglected. An analytical-Ritz method is developed to study the vibratory characteristics of the plate in contact with the fluid. First of all, the exact expression of the motion of the fluid is obtained, in which the unknown coefficients are determined by using the method of separation of variables and the method of Fourier series expansion. Then, the Ritz approach is used to obtain the frequency equation of the system. The vibrating beam functions are adopted as the admissible functions for the wet-mode expansion of the plate, and the added virtual mass incremental (AVMI) matrices are obtained for plates with arbitrary boundary conditions. Finally, a convergence study is carried out and some numerical results are given. The accuracy of AVMI factor solutions is discussed by comparing with the more accurate analytical-Ritz solutions presented in this paper. Furthermore, It is seen that the present method is also suitable for the vibration analysis of rectangular plates in contact with infinite fluid by taking the finite, but larger size fluid domain as an approximation in the computation.     

Plotnikov——Toland板模型中水弹性孤立波的迎撞

通过奇异摄动方法研究了在薄冰层覆盖的不可压缩理想流体表面上传播的两个水弹性孤立波之间的迎面碰撞.借助特殊的 Cosserat 超弹性壳 理论以及Kirchhoff--Love 板理论,冰层由 Plotnikov--Toland板模型描述.流体运动采用浅水假设和Boussinesq 近似. 应用Poincaré--Lighthill--Kuo 方法进行坐标变形,进而渐近求解控制方程及边界条件, 给出了三阶解的显式表达. 可以观察到碰撞后的孤立波不会改变它们的形状和振幅. 波浪轮廓在碰撞之前是对称的, 而在碰撞之后变成不对称的并且在波传播方向上向后倾斜. 弹性板和流体表面张力减小了波幅. 图示比 较了本文与已有结果可知线性板模型可作为本文的一个特例.      

Optimum material design for functionally gradient material plate

Summary Steady thermal stresses in a plate made of a functionally gradient material (FGM) are analyzed theoretically and calculated numerically. An FGM plate composed of PSZ and Ti-6Al-4V is examined, and the temperature dependence of the material properties is considered. A local safety factor is used for evaluation of the FGM's strength. It is assumed that top and bottom surfaces of the plate are heated and kept at constant thermal boundary conditions. The pairs of the surface temperatures, for which the minimum local safety factor can be of more than one, are obtained as available temperature regions. The temperature dependence of the material properties diminishes, available temperature region as compared with that for an FGM plate without it. The available temperature region of the FGM plate is wider than that of the two-layered plate, especially for the surface temperatures which are high at the ceramic surface and low at the metal side. The influence of different mechanical boundary conditions is examined, and available temperature regions are found to be different, depending on the mechanical boundary conditions. The influence of the intermediate composition on the thermal stress reduction is also investigated in detail for the surface temperatures which are kept at 1300 K at the ceramic surface and 300K at the metal side. Appropriate intermediate composition of the FGM plate can yield the local safety factor of one or more for the four mechanical boundary conditions at once. For the two-layered plate there does not exist, however, any appropriate pair of metal and ceramic thicknesses which would yield the local safety factor of one or more for the four mechanical boundary conditions at once. The influence of the intermediate composition on the maximization of the minimum stress ratio depends on the mechanical boundary conditions. Finally, the optimal FGM plates are determined.