Thin Flat Plate Research Articles

The sudden melting of a thin vertical flat plate in a Darcian free convection flow

In this paper a solution of the unsteady free convection boundary-layer equations in porous media is presented for the sudden melting of a thin semi-infinite vertical flat plate. It is assumed that fort<0 that the steady state temperature and velocity has been obtained and at timet=0 the plate starts to melt. Three distinct phases of the temporal development of the flow are considered. An analytical solution is presented which is valid fort≪1 and which shows the initial decay of the boundary-layer. At very large values oft the boundary-layer will disappear and an analytical solution showing this approach is obtained. A numerical solution which matches these two limiting solutions has been obtained by means of a step-by-step method and these results show good agreement with the theoretical results that are valid at small and large values of time.

A basic study of the accuracy estimation of the finite element method. (2nd Rort, Analysis of transverse vibration of thin flat plates).

As a basic study for the establishment of an accuracy estimation method in the finite element method, this report deals with the problems of free transverse vibration of thin and flat plates. From the numerical experiments for a uniform mesh division, the following relation was deduced, e∝(h/a)k, kl, where e is the error of the value by the finite element method relative to the exact value and h/a is the dimensionless mesh diameter. Using this relation, an accuracy estimation method, which was based on the recursive simulation method, is presented. A computer program using this accuracy estimation method is developed and applied to many problems with various shapes. The usefulness of this accuracy estimation method is illustrated by these application results.

Prediction of loaded airfoil unsteady aerodynamic gust response by a locally analytical method

A complete first-order model is formulated to analyze the effects of steady loading on the incompressible unsteady aerodynamics generated by a two-dimensional gust convected with the steady mean flow past an arbitrary airfoil at finite nonzero angle of attack. A locally analytical solution is then developed in which the discrete algebraic equations which represent the flow field equations are obtained from analytical solutions in individual grid elements. The unsteady flow field is rotational and is linearized about the full potential steady flow past the airfoil. Thus, the effects of airfoil geometry and angle of attack are completely accounted for through the mean potential flow field. The steady flow is independent of the unsteady flow. However, the strong dependence of the unsteady flow on the steady effects of airfoil geometry and finite angle of attack are manifested in the unsteady boundary conditions which are coupled to the steady flow. A body-fitted computational grid is utilized. Analytical solutions to the transformed flow equations in individual grid elements are then developed, with the complete solution obtained by assembling these locally analytical solutions. This model and locally analytical solution are then applied to a series of airfoil and flow configurations. The results demonstrate that accurate predictions for the unsteady aerodynamic gust response are obtained only by including the coupled steady flow effects on the unsteady aerodynamics. Thus for cambered, or cambered and thick airfoils at zero or finite angle of attack, or a thin flat plate airfoil at a nonzero angle of attack, the model and solution developed herein accurately predict the gust response. It was also demonstrated that the classical small perturbation combined transverse and chordwise gust models yield accurate predictions only for the special case of a thin flat plate airfoil at zero angle of attack, i.e. only when the chordwise gust is zero.

Buckling, postbuckling, and nonlinear vibrations of imperfect plates

Formulations and computational procedures are presented for studying the geometrically nonlinear behavior, including buckling, postbuckling, and nonlinear vibrations of perfect and imperfect, isotropic and laminated thin plates. The finite-element method is used. The element used is a 48 degree-of-freedom thin flat plate rectangular element capable of modeling arbitrary imperfections. The incremental and total stiffness matrices for large displacement behavior are derived based on the total Lagrangian approach in conjunction with the Hamilton's principle. The geometric imperfections are treated by considering additional terms in the straindisplacement relations. Numerical results are presented for a variety of examples including: 1) postbuckling analysis of an isotropic, imperfect flat plate; 2) postbuckling analysis of a thin, cross-ply laminated imperfect plate; 3) free vibrations of an angle-ply laminated plate; 4) free-vibrations of an imperfect isotropic plate under the action of axial loads; 5) nonlinear free-vibrations of isotropic perfect flat plates; 6) nonlinear vibrations of axially loaded, isotropic perfect flat plates; and 7) nonlinear vibrations of an imperfect laminated plate. The results obtained in this study are compared with existing solutions and a good agreement is seen.

Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.

A basic study of the accuracy estimation of structural analysis by the zooming method. (2nd report. Finite elemnt analysis of the transverse bending of thin flat plates. (Part 2)).

As a basic study for the establishment of an accuracy estimation method in structural analysis using the zooming method, this second report deals also, as previous report, with the finite element analysis of the problems of transverse bending of thin, flat plates. In the previous report, we treated a small hole, if it was in a plate, as a hole. This treatment inconveniently led to an increase of nodal points and hence to an increase of the size of the system of simultaneous linear equations and computation time. To overcome this inconvenience, we propose in this second report a method where small holes are neglected and the whole region is divided into elements of nearly equal size at the initial two or more analyses prior to the start of the zooming process. But approximation error due to the neglect of small holes must be added to the error of the zooming method with consideration of holes. A computer program using this proposed method was developed and applied to several problems. The usefulness of this method was illustrated by these application results.

A basic study of the accuracy estimation of structural analysis by the zooming method. (1st report Finite element analysis of the transverse bending of thin flat plates)

As a basic study for the establishment of an accuracy estimation method in structural analysis using the zooming method, this report deals with the finite element analysis of the problems of transverse bending of thin, flat plates. From numerical experiments for uniform mesh divisions, the following relation was deduced, e∝(h/a)k, kg-1, where e is the error of the value by the finite element method relative to exact value and h/a is the dimensionless mesh diameter. Using this relation, the errors of gradients on a zooming boundary and that of the bending moment at the selected points in the zooming region can be estimated by the recursive analyses of the whole region or of the zooming region. The error of the bending moment at the selected points is expected to be smaller than the sum of these two errors. A computer program using this error estimation method was developed and applied to sixteen problems of various shapes. The usefulness of this accuracy estimation method was illustrated by these application results.

A basic study of the accuracy estimation of structural analysis by the zooming method. Finite element analysis of the transverse bending of thin flat plates.

As a basic study for the establishment of an accuracy estimation method in structural analysis using the zooming method, this report deals with the finite element analysis of the problems of transverse bending of thin, flat plates. From numerical experiments for uniform mesh divisions, the following expression was deduced, ε∝(h/a)k, k≥1, where ε is the error of the value by the finite element method relative to the exact value, and h/a is the dimensionless mesh diameter. Using this relations, the errors of gradients on a zooming boundary, and that of the bending moment at selected points in the zooming region can be estimated by the recursive analyses of the whole region or of the zooming region. The error of the bending moment at the selected points is expected to be smaller than the sum of these two errors. A computer program using this errors estimation method was developed and applied to sixteen problems of various shapes. The usefulness of this accuracy estimation method was illustrated by these application results.

Analysis of Turbulent Flow Past a Class of Semi-Infinite Bodies

This study was undertaken with the primary purpose of developing an analysis for flow past a class of two-dimensional and axisymmetric semi-infinite bodies. The time-averaged Navier-Stokes equations for these flows are derived in surface-oriented conformal coordinates (ξ, η) in terms of similarity-type vorticity and stream-function variables. Turbulence closure is achieved by means of a two-equation turbulence model utilizing the kinetic energy k and its dissipation rate ε which enable determination of the isotropic eddy viscosity. The coupled vorticity and stream-function equations are solved simultaneously using an incremental formulation of the factored alternating-direction implicit scheme. The turbulence equations for k and ε are solved by the standard ADI method. Numerical solutions are obtained for the thin flat plate and compared with available experimental and analytical data. Also, results are obtained for flow over a parabola and compared with the flat-plate results in order to assess the effects of longitudinal curvature on the flow results. Finally, solutions are obtained for flow past a two-dimensional semi-infinite body with a shoulder, at Red = 24,000. The computed results have the same general trend as the experimental data; possible causes for the differences within the separated-flow region are cited.

Turbulence structure in the intermediate wake of a rotating flat plate. (Large-scale vortices)

The structure of large-scale vortices behind a thin flat plate being rotated in a uniform stream around its axis with a constant angular velocity is described experimentally by a flow-visualization technique and a phase-averaging technique. The time-mean velocity, turbulent energies and Reynolds shear stresses are also obtained. These measurements are made with 712 plate widths downstream of the centre of the plate; this region was termed as the intermediate wake. Using the phase of rotation of the plate as a reference signal, phase-averaging is made with a minimal dispersion because the large-scale vortices in the intermediate wake were well locked to the rotation. Results are presented for a Reynolds number of 1.25 × 104 and for spin parameters of S=0.28±0.003 and 0.51±0.004. The structure and motion of the large-scale vortices are demonstrated in terms of the vorticity contours in a space-time (or space-phase) domain. a wide region of the negative turbulent-energy production is found in the wake.

A Basic Study of the Accuracy Estimation of the Finite Element Method : 1st Report, Analysis of Transverse Bending of Thin Flat Plate

As a basic study for the establishment of an accuracy estimation method in the finite element method, this report deals with the problems of transverse bending of thin flat plates. From numerical experiments for uniform mesh divisions, the following relation was deduced; e∝(h/a)k, kg1, whereeis the error of the value by the finite element method relative to the exact value and h/a is a dimensionless mesh diameter. Combining this relation with the results obtained by error analysis for local mesh refinement, an accuracy estimation method based on the recursive simulation method was presented. A computer program using this accuracy estimation method was developed and applied to nineteen problems of various plate shapes. The usefulness of this accuracy estimation method was illustrated by these application results.

A basic study of the accuracy estimation of the finite element method (1st report, Analysis of transverse bending of a thin flat plate)

As a basic study for the establishment of an accuracy estimation method in the finite element method, this report deals with the problems of transverse bending of thin, flat plates. From numerical experiments for uniform mesh divisions, the following relation was deduced, e∝(h/a)k, kg1, where e is the error of the value by the finite element method relative to the exact value and h/a is the dimensionless mesh diameter. combining this relation with the results of the error analyses for local mesh refinement, an accuracy estimation method, which was based on the recursive simulation method, was presented. A computer program using this accuracy estimation method was developed and applied to nineteen problems of various shapes. The usefulness of this accuracy estimation method was illustrated by these application results.

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed 1 doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and ũ (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C 1 displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.

Edge waves in a thermoelastic plate

Temperature-rate-dependent thermoelasticity theory is employed to study waves propagating along the edges of a thin flat plate of infinite length, which is in a state of plane stress. Governing equations of the plane stress problem are derived and it is found that the speed of heat waves in this problem is, in general, less than that in the plane strain and the general three-dimensional problems. The waves of the desired type are analysed for isothermal and insulated edge conditions, and the paths of particles as well as the cut-off frequencies are determined. Waves in a thin bar and in a semi-infinite plate are discussed as limiting cases. The effects of the finiteness of the neat propagation speed are analysed in some detail.

Two-Dimensional Mathematical Model of Evacuated Tubular Solar Collector

Analysis of an evacuated tubular solar collector is presented by developing a two-dimensional performance model. The collector uses a thin flat plate spanning its diameter as its absorbing surface. Energy balances are made on collector plate and tube, each considered as a separate unit. It has been found that a zero capacitance model is quite adequate when hourly meteorological data are used, and hence in this study steady state analysis of the collector is made [1]. The overall loss coefficient has been assumed to be constant for the whole plate. The model also includes optical effects. The emphasis of the investigation is to study the two-dimensional effects in the absorber plate housed in evacuated glass cylinders. Performance curves obtained from the two-dimensional model has been compared with that of HW model. It was found that the HW model overestimates the performance.

Recombination kinetics of OH radicals on a quartz wall in a propaneoxygen flame at 25 Torr

The boundary layer between a thin flat quartz plate and a propaneoxygen premixed flame at 25 Torr has been studied, under conditions where the flow is parallel to the surface of the plate. Profiles of velocity, temperature, and OH concentration have been determined. First order recombination kinetics of OH on the quartz wall are determined from these data by three different procedures: the gradient method, the integral method, and through Rosner's theory. An extension of Rosner's theory has been developed for the last of these three methods.

The centrosome cycle in the mitotic cycle of sea urchin eggs

When sea urchin eggs entering mitosis are exposed to an appropriate concentration of mercaptoethanol, the chromosome cycle is restrained while the centrosome cycle advances. The two poles of the mitotic apparatus separate into four poles, while the chromosomes remain in their metaphase arrangements until released by the removal of the mercaptoethanol. We follow the centrosomes through the stages of the generation of two poles by each original pole. In electron microscopic studies, the osmiophilic component of the centrosomes serves as an indicator of their changing forms as each pole generates two poles. In light microscopic studies, including observations of birefringence, the shapes of the polar ends of the spindles are taken as indicators of the shapes of the centrosomes. The successive stages of the centrosome cycle are (1) compact spherical centrosomes at the time of formation of the mitotic apparatus; (2) expansion and flattening of the centrosomes, leading to (3) formation of thin flat plates, perpendicular to the spindle axis. Corresponding to the extended flat shape of the centrosomes, the spindle poles are flat; microtubules ‘point’ to the centrosomal plate and not the centrioles. The centrioles are separated in the flattening of the centrosomes. (4) The flat plate divides into two and each of the two halves becomes more compact, defining two separate poles. Our findings resurrect and update Boveri's [5] observations and interpretations of the centrosome. Centrosomes have shapes. The shapes may be imparted to the microtubular structures that they generate. The formation of two separate centrosomes from one, in the formation of mitotic poles, is describable as a sequence of changes in shape.

A model for polarization effects in remote probing of clouds

A computer model is described which has been developed to aid in the interpretation of polarization data obtained by electromagnetic remote probing of mixed phase ice crystals and waterdrop clouds. Such clouds can contain many possible crystal forms, most notably thin long cylinders, bullets, and flat plate crystals, and the intent of the measurements is to characterize the ice crystal population by the polarization changes caused by these highly nonspherical shapes. The present work is a partial investigation of the theoretical aspects of this problem using flat circular disks as models of flat plate ice crystals randomly oriented over all rotation angles about axes perpendicular to the flat faces of the crystals. Mixtures of such disks and spherical particles to simulate waterdrops are considered. It is expected to include other crystal shapes in future work. Mie theory is used for the waterdrops, which are modeled as spheres, and a numerical procedure especially designed for dielectric disks is used for the flat plate crystals. Numerical results are presented to illustrate the sensitivity of the different Stokes parameters to ice crystal size and orientation distribution and variables such as the relative number density of ice crystals to waterdrops.

Transport in the near aerodynamic wakes of flat plates

An experimental investigation has been carried out into the nature of the transport of airborne material in the near aerodynamic wakes of bluff bodies with simple shapes. The main attention was focused on the essential differences existing between axi- symmetric flows (as about disks) and two-dimensional flows (as about rectangular long thin flat plates). Measurements were made for such bodies of the near-wake residence time of injected small particles, along with other and more familiar near- wake properties such as the vortex-shedding frequency and base pressure. It was concluded for disks that the transport of material into and out of the near-wake region is dominated by turbulent diffusion, and is strongly influenced by free-stream turbulence, especially for free-stream turbulence whose length scale is substantially smaller than the disk diameter. For rectangular flat plates, transport is dominated by the periodic shedding of vortices, and to only a secondary extent by turbulent motions, and is not strongly influenced by free-stream turbulence.

Free Convection Heat Transfer near Leading Edge of Semi-infinite Vertical Flat Plate : 2nd Report, Uniform Heat Generation, Pr=0.72

Free convection heat transfer near the leading edge of semi-infinite vertical thin flat plate with uniform heat generation, has been analyzed numerically by the finite-difference method for Pr=0.72. For uniform heat flux case, the present Nusselt number distribution of free convection heat transfer is approximated well by the following equation : [numerical formula] The surface temperature distribution near the leading edge of the flat plate with uniform heat generation, which is placed vertically in air, has been calculated by considering not only free convection heat transfer but also thermal radiation and axial heat conduction in the plate. The effects of axial heat conduction in the plate on the surface temperature distribution near the leading edge are expressed in terms of the dimensionless parameter KD. The calculated temperature distributions agreed well with the experimental results using a Mach-Zehnder interferometer.