Oscillations of a plate cascade in a transonic gas flow

The transonic unsteady flow of a gas through a cascade of thin, slightly curved plates is quite complex and has received little study. The main difficulties are associated with the nonlinear dependence of the aerodynamic characteristics on the plate thickness. In [1] it is shown that, for a single thin plate performing high-frequency oscillations in a transonic gas stream, the variation of the unsteady aerodynamic characteristics with plate thickness may be neglected. For a plate cascade, the flow pattern is complicated by the aerodynamic interference between the plates, which may depend significantly on their shape. A solution of the problem of transonic flow past a cascade without account for the plate thickness has been obtained by Hamamoto [2].The objective of the present study is the clarification of the dependence of the aerodynamic characteristics of a plate cascade on plate thickness in transonic unsteady flow regimes. The nonlinear equation for the velocity potential is linearized under the assumption that the motionless plate causes significantly greater disturbances in the stream than those due to the oscillations. A similar linearization was carried out for a single plate in [3]. The aerodynamic interference between the plates is determined by the method presented in [4]. As an example, the aerodynamic forces acting on a plate oscillating in a duct and in a free jet are calculated.     

Repair of a Plate with a Circular Hole by Applying a Patch

The stressed state of a thin elastic infinite plate with a circular hole covered by a circular patch of a greater radius is considered. The center of the hole coincides with the center of the patch. The patch is attached to the plate along its entire boundary. Stresses are prescribed at infinity on the plate and at the hole boundary. Complex Muskhelishvili potentials are found by the method of power series, and the behavior of stresses on the patch–plate interface and at the hole boundary is studied.     

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.     

Impact of a rectangular plate on a liquid half-space

The potential flow of an ideal incompressible fluid occupying a half-space resulting from the impact of a rectangular plate on its surface is considered. Outside the plate the surface of the fluid is free. An integral equation of the first kind is obtained for the impulsive pressure beneath a flexible plate. It is solved on a computer by the power series method for the particular case of a rigid nondeformable plate. The accuracy of the method is estimated. The theoretical dependence of the virtual mass and virtual moment of inertia coefficients of a rigid nondeformable plate on the plate geometry is constructed and compared with the experimental data and with empirical formulas [1-3] not directly related with the solution of the Laplace equation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 120–126, September–October, 1992.     

Oblique impact by a spherical solid on a metal plate

The results of experimental studies of oblique impact by a spherical solid on a plane metal plate are presented. Plates made of AMTsM (plastic) and D16AT (rigid) alloys were used to determine the influence of the material characteristics on the impact aftereffects. The dependence of the ballistic limit on the angle of the impact velocity vector inclination to the plate plane was determined. The relationship between the ballistic limits of the oblique and normal impacts on the plate is established. The dependence of the ball motion trajectory refraction angles after collision with the plate on the ratio of the initial velocity to the ballistic impact is determined.     

Elastoplastic analysis of a bent circular plate with a central hole

A circular aluminum plate with a small concentric hole (1/10 the plate thickness) and supported on its outer edge by a ring was subjected to a concentrated load at its center, applied through a rigid ball of radius equal to the plate thickness. Strains were determined using grids, moiré, and electrical strain gages on the top and bottom surfaces of the plate for loads up to and including the one associated with the appearance of the first crack in the plate. The investigation is related to the development of specimens to be used to determine fracture characteristics of materials used in lightweight construction.     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.     

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.     

Stokes flow due to the sliding of a smooth plate over a slotted plate

The Stokes flow due to the sliding of a smooth plate over a slotted plate and a bottom fixed plate is solved by eigenfunction expansions and matching. The streamlines show recirculating bubbles near the slots. The drag depends on the void ratio and the relative positions of the three plates. It is found that the Navier condition is in general not satisfied for a slotted plate.     

Wave action on a movable plate immersed into a fluid. Physical and numerical experiments

The effect of the motion of a rigid inclined plate immersed into a fluid counter incident waves is studied both experimentally and numerically. In the wave water channel experiments the velocity of a trolley with a plate running freely on rails is determined as a function of the wave parameters, the immersion depth, the angle of inclination, and the plate dimensions. The interaction between the traveling waves and the plate having a single translational degree of freedom along the horizontal axis is numerically calculated in the time-dependent, two-dimensional formulation. The dependence of the upwave motion effect on the parameters varied in the full-scale experiment is analyzed. In the numerical experiment a regime of the downwave plate motion at a constant high velocity is found to exist. The channel bottom effect is estimated and the behavior of the plate with a flap is studied.     

Unsteady motion of a source in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal straight motion of a point source of mass in a fluid of infinite depth. The solution is carried out using known integral and asymptotic methods. The formulas obtained are used to numerically analyze the effect of plate thickness, depth of submergence of the source, and its acceleration, deceleration, and velocity of uniform motion on the deflection of the floating plate.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.     

The flexibility effect of a plate according to various angles of attack in a free-stream

This paper presents a computational fluid–structure interaction analysis for a flexible plate in a free-stream to investigate the effects of flexibility and angle of attack on force generation. A Lattice Boltzmann Method with an immersed boundary technique using a direct forcing scheme model of the fluid is coupled to a finite element model with rectangular bending elements. We investigated the effects of various angles of attack of a flexible plate fixed at one of the end edges in a free-stream at a Reynolds number of 5000, which represents the wing flapping condition of insects and small birds in nature. The lift of the flexible plate is maintained at the large angle of attack, whereas the rigid plate shows the largest lift at angles of attack around 30–40° and then drastic reductions in the lift at the large angle of attack. If we consider the efficiency as the lift divided by the drag, the flexible plate shows better efficiency at angles of attack greater than 30° compared to the rigid plate. The better performance of the flexible plate at large angles of attack comes from the deformation of the plate, which produces an interaction between the trailing edge vortex and the short edge vortex. The horseshoe-shaped vortex produced by a large vortex interaction at the trailing edge side has an important role in increasing the lift, and the small projection area due to the deformation reduces the drag. Furthermore, we investigate the role of flexibility on the lift and the drag force of the rectangular plate in a free-stream as the Reynolds number increases. Whenever a large vortex interaction at the trailing edge side is shown, the efficiency of the rectangular plate is improved. Especially, the flexible plate shows better efficiency as the Reynolds number increases regardless of the angle of attack.     

On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

We prove the duality of solutions for the problem of determining the boundary conditions on two opposite sides of a rectangular plate from the frequency spectrum of its bending vibrations. A method for determining the boundary conditions on two opposite sides of a rectangular plate from nine natural frequencies is obtained. The results of numerical experiments justifying the theoretical conclusions of the paper are presented. Rectangular plates are widely used in various technical fields. They serve as printed circuit boards and header plates, bridging plates, aircraft and ship skin, and parts of various mechanical structures [1–4]. If the plate fixing cannot be inspected visually, then one can use the natural bending vibration frequencies to find faults in the plate fixing. For circular and annular plates, methods for testing the plate fixing were found in [5–7], where it was shown that the type of fixing of a circular or annular plate can be determined uniquely from the natural bending vibration frequencies. The following question arises: Is it possible to determine the type of fixing of a rectangular plate on two opposite sides of the plate from the natural bending vibration frequencies if the other two sides are simply supported? Since the opposite sides of the plate are equivalent to each other, a plate with “rigid restraint—free edge” fixing will sound exactly the same as a plate with “free edge—rigid restraint” fixing. Hence we cannot say that the type of fixing of a rectangular plate on two opposite sides can be uniquely determined from its natural bending vibration frequencies. But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs.     

Wave-induced dynamics of a particle on a thin circular plate

In this paper, the dynamics of a particle placed on a thin circular plate carrying circumferential harmonic travelling wave is studied. Coulomb friction is used to model the particle–surface interaction. Distinct regions on the plate surface are identified where either of the three phases of particle motion, namely jumping, sliding and sticking, occurs. Also, the effect of wave frequency and the plate geometry on these regions is studied. Interestingly, there exists an optimum plate thickness for which the region of sliding is maximum. At certain wave frequencies, from the numerical simulations within sticking and sliding regions, it is observed that the average particle motion spirals inwards towards the plate centre. Such an average motion is observed whenever the particle is placed initially with a zero velocity relative to the plate surface. The Gedanken experiments discussed herein provide cogent explanations to all the observed average (slow) dynamics and are also found to be useful in predicting the slow dynamics of the particle a priori, that is, before the actual numerical simulations. The particle’s velocity couples the radial and tangential sliding friction components and is found to be the key physical feature that explains the observed behaviour. Also, it is observed that the plate surface excited by circumferential travelling waves can provide acoustic lubrication to a particle by reducing the limiting force required to move it relative to the surface. The methods discussed in this paper can be extended to study the dynamics of a group of particles (granular materials) and extended rigid bodies, interacting with such surface waves.

     

Experimental heat transfer from a vertical plate within a naturally ventilated cabinet

 The paper deals with the experimental investigation of heat transfer from a vertical isothermal plate mounted inside a cabinet with lateral openings. Ambient air is drawn into the cabinet and cools the plate by natural convection. The experiments were conducted by varying the distance between the plate and the cabinet walls, the plate-to-ambient temperature difference and the size, position, shape and number of the vent openings. Local and average heat transfer coefficients along the plate were measured by the schlieren optical technique. Heat transfer rates from the plate turned out to be significantly influenced by the geometric and thermal parameters investigated. The results provide a useful tool for thermal design of cooling systems by air natural ventilation. Received on 17 January 2000     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

Unsteady radiative-convective heat transfer in a high-temperature gas-particle flow past a semi-transparent plate

Numerical simulations of unsteady radiative-convective heat transfer in a turbulent flow of a mixture of gases and solid particles past a semi-transparent plate are performed. An ablation process is demonstrated to occur on the plate surface in the case of intense radiative heating of the plate by an external source with emission in a limited spectral range. Temperature fields and distributions of heat fluxes in the boundary layer and in the plate are calculated. Calculation results are presented, which allow determining the effect of ablation and reflecting properties of the plate surface on the thermal state of the medium in the system containing the boundary layer and the plate under conditions of plate heating by a high-temperature source of radiation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 140–146, May–June, 2009.     

Interaction of a rigid punch and a circular plate with a fixed side and a stress-free face

The axisymmetric contact problem of a rigid punch indentation into an elastic circular plate with a fixed side and a stress-free face is considered. The problem is solved by a method developed for finite bodies which is based on the properties of a biorthogonal system of vector functions. The problem is reduced to a Volterra integral equation (IE) of the first kind for the contract pressure function and to a system of two Volterra IE of the first kind for functions describing the derivative of the displacement of the plate upper surface outside the punch and the normal (or tangential) stress on the plate lower fixed surface. The last two functions are sought as the sum of a trigonometric series and a power-law function with a root singularity. The obtained ill-conditioned systems of linear algebraic equations are regularized by introducing small parameters and have a stable solution. A method for solving the Volterra IE is given. The contact pressure functions, the normal and tangential stresses on the plate fixed surface, and the dimensionless indentation force are found. Several examples of a plane punch computation are given.