Flat Plate Solution Research Articles

Investigation for Stress Intensity Factor Solutions on Nozzle Corner Flaw (Part 2)

In order to provide a stress intensity factor (K)-solution for a nozzle corner flaw subjected to internal pressure for the Japan Society of Mechanical Engineers (JSME) Rules on Fitness for Service (FFS) for nuclear power plant, four candidates, which are the Paris and Sih based solution, the Fife’s solution, the Kobayashi’s solution and the solution of a flat plate with a semi-elliptical surface flaw in the JSME Rules on FFS, were compared with the Ks by Finite Element Method. As the result, the flat plate solution was confirmed to be the most adequate for K calculation of a nozzle corner flaw in the JSME Rules on FFS.

Analytical solution for an acoustic boundary layer around an oscillating rigid sphere

Analytical solutions in fluid dynamics can be used to elucidate the physics of complex flows and to serve as test cases for numerical models. In this work, we present the analytical solution for the acoustic boundary layer that develops around a rigid sphere executing small amplitude harmonic rectilinear motion in a compressible fluid. The mathematical framework that describes the primary flow is identical to that of wave propagation in linearly elastic solids, with the difference being the appearance of complex instead of real valued wave numbers. The solution reverts to the well-known classical solutions in special limits: the potential flow solution in the thin boundary layer limit, the oscillatory flat plate solution in the limit of large sphere radius, and the Stokes flow solutions in the incompressible limit of infinite sound speed. As a companion analytical result, the steady second order acoustic streaming flow is obtained. This streaming flow is driven by the Reynolds stress tensor that arises from the axisymmetric first order primary flow around such a rigid sphere. These results are obtained with a linearization of the non-linear Navier–Stokes equations valid for small amplitude oscillations of the sphere. The streaming flow obeys a time-averaged Stokes equation with a body force given by the Nyborg model in which the above-mentioned primary flow in a compressible Newtonian fluid is used to estimate the time-averaged body force. Numerical results are presented to explore different regimes of the complex transverse and longitudinal wave numbers that characterize the primary flow.

Viscous fluid flow inside an oscillating cylinder and its extension to Stokes’ second problem

We present the analytical solution for the fluid motion inside a cylindrical tank whose angular velocity starts from rest and undergoes a harmonic oscillation. This problem, which has not yet been reported, is an extension to Stokes’ second problem where the fluid motion is governed by an outer moving cylindrical boundary and a zero velocity condition at the cylinder center. Different from the flow on the outside of a cylinder, the cylinder radius has a large influence on the internal fluid motion. We show that the fluid approaches solid body rotation for cylinders with outer radii similar to the characteristic viscous length scale of the flow, whereas the motion approaches that of Stokes’ original flat plate solution within very large cylinders. We detail both the transient starting condition and the quasi-steady fluid motion, which we present along with a particle image velocimetry experiment for validation. After decay of the initial startup transient, both quasi-steady analytical and experimental results predict that the oscillatory flow inside has an amplitude of velocity that decreases toward the center of the cylinder. The thickness of the Stokes layer, which is proportional to the penetration depth of the viscous wave, is altered by the size of the cylinder and/or the frequency of oscillation. We show that the penetration depth of the Stokes layer reaches its maximum thickness at intermediate cylinder sizes. The solution and results presented herein are potentially of value to describe the fluid motion in many applications where fluids are contained within cylindrical geometries.

Boundary layer measurements over a body of revolution using long-distance particle image velocimetry

This study reports the development and application of long-distance high magnification Particle Image Velocimetry (PIV) to measure the mean and fluctuating velocity components in a turbulent boundary layer over a body of revolution. A 400 mm lens coupled with an extension tube and a novel aperture control mechanism has enabled high resolution images at a working distance of 1.5 m. The body of revolution used in the investigation is representative of a conventional submarine hull without appendages, as per (Joubert, 2006). The flow over the model was measured at a nominal Reynolds number, ReL=4.0 × 106, based on the length of the body. A comparison of single-point statistics from the PIV experiments were found to compare well with pitot probe and hot-wire measurements. Further, the boundary layer thickness and mean streamwise velocity were found to conform closely with a traditional flat plate turbulent boundary layer in the parallel mid-body section of the geometry. However, a comparison of the streamwise and wall-normal Reynolds stresses with Direct Numerical Simulations (DNS) of a turbulent boundary layer on a flat plate with zero pressure gradient indicated lower turbulence levels. This was partly attributed to the spatial averaging evident both in the hot-wire and PIV measurements. We also report that in the tail section of the model, the boundary layer rapidly grows and departs from the flat plate solution due to the conical section and adverse pressure gradient imposed on the flow.

Stress magnification factor for angular misalignment between plates with welding-induced curvature

The construction of lightweight structures poses new design challenges as a result of the different mechanics of deformation experienced by thinner-plated structures. Because of a reduced bending stiffness, thin plates are particularly sensitive to welding-induced distortions, which include a curvature, in addition to the axial and global angular misalignment observed on thick plates. The curvature shape and amplitude determine a local angular misalignment at the welded joint, which causes non-negligible secondary bending effects. Therefore, the commonly used stress magnification factors km solution for flat plates need a further development to include the curvature effect. This study proposes new analytical formulations, which extend the applicability of the existing solutions to the assessment of the structural stress of a curved thin plate under an axial load. The improved formulations are consistent with the geometrical non-linear finite element analysis under compression (up to sim 80% of the buckling limit) and tension (up to the yield strength). A sensitivity analysis is presented in order to show the dominant role of the curvature effect in the estimation of the km factor. Regardless of the load applied, the presence of the curvature causes a flat plate solution inaccuracy greater than 10% when the local angular misalignment is more than 1.25 times higher than the global angular misalignment in the case of a thin and slender structure.

An analytical correction to Amiet’s solution of airfoil leading-edge noise in non-uniform mean flows

Gust/turbulence–leading edge interaction is a significant source of airfoil broadband noise. An approach often used to predict the sound is based on Amiet’s flat-plate solution. Analytical studies have been conducted to investigate the influences of airfoil geometries, non-uniform mean flows and turbulence statistics, which, however, were often too convoluted. In this work, the problem is revisited by proposing simple corrections to the standard flat-plate solution to account for the effect of non-uniform mean flows of real airfoils. A key step in the method is to use a new space–time transformation that is analogous to the Prandtl–Glauert transformation to simplify the sound governing equation with spatially varying coefficients to a classical wave equation, which is then solved using the Schwarzschild technique as in Amiet’s solution. The impacts of Mach number, wavenumber and airfoil geometry on the prediction accuracy are investigated for both single-frequency and broadband cases, and the results are compared against high-fidelity simulations. It predicts the sound reduction by the airfoil thickness, and reveals that the reduction is caused by the non-uniform streamwise velocity. The limitations of the model are discussed and the approximation errors are estimated. In general, the prediction error increases with the airfoil thickness, the sound frequency and the flow Mach number. Nevertheless, in all cases studied in this work, the proposed correction can effectively improve the prediction accuracy of the flat-plate solution much more efficiently compared to numerical solutions of the Euler equations using computational aeroacoustics.

On the effect of streamwise disturbance on the airfoil-turbulence interaction noise.

Usually the prediction of airfoil-gust interaction noise is mainly based on the flat plate solution, and only the transverse velocity disturbance that is perpendicular to the flat plate surface is considered. However, the solution might underestimate the sound generation in non-uniform mean flows for finite thickness airfoils, in which the velocity disturbance in the streamwise direction could probably produce sound as well. The effect is studied in this work by using both numerical simulations and theoretical analysis. To highlight the effect of streamwise disturbance, the anisotropic turbulences with constant transverse velocity components while varying streamwise components are employed, which can be realised by adjusting the integral length scales in the Gaussian energy spectra. The simulation results explicitly show that more sound is produced by the streamwise components for airfoils at higher Mach numbers when the flow non-uniformities are more significant. An analytical model based on acoustic analogy to account for the sound generation by the coupling between the gust and non-uniform mean flows is proposed, which explains the observations in the numerical simulation well. Also, a correction to Amiet's flat plate solution using the proposed model yields better agreement with the numerical results, especially for anisotropic turbulences with larger integral length scale in the streamwise direction.

Numerical study on girth weld of marine steel tubular piles

At a splice weld of marine steel tubular pile, the misalignments between two adjacent pile segments may cause significant stress concentrations. In structural analysis, the stress concentrations shall be properly addressed, normally by a stress concentration factor (SCF) in practice. Based on a flat-plate configuration, the SCF at pipe splice under either axial tension or in-plane bending moment has been theoretically derived. To verify the effectiveness of the flat-plate solutions, this paper investigated the SCFs with numerical modeling. Finite element models built by ANSYS were used to simulate pipe splices for different pile diameters and wall thicknesses, which are representative of practical marine applications. Axial tension and in-plane bending moment, as well as their combination were applied. The flat-plate solutions were compared with the numerical results. The results show that the flat-plate solutions are close to the numerical results, indicating that they are reasonably effective in practical applications under complex loading conditions. The findings will significantly help in the hot-spot analysis for splice welds of steel tubular members, particularly piles, in marine structures given complex loading effects. Additionally, an integrated formula including the effect of both axial tension and in-plane bending moment is formulated for the SCF at pile splice.

Verification and validation of Reynolds-averaged Navier–Stokes turbulence models for external flow

The Spalart–Allmaras (S-A) turbulence model in the NASA-Langley CFL3D and FUN3D flow solvers has been previously verified 2nd-order accurate. For low subsonic 2-D applications (turbulent flat plate and NACA 0012 airfoil at α=0°), solutions from the S-A, S-A with Rotation and Curvature (SARC), Menter Shear-Stress Transport (SST), and Wilcox 1998 k-ω turbulence models in commercial flow solvers, Cobalt and RavenCFD, are compared with NASA results for code verification. Of 36 case evaluations, each of which uses 5 systematically refined computational meshes, only 7 approach 2nd-order observed accuracy, but 27 cases show 1st-order or better, indicating the formal order may be less than 2 for these applications. Since Cobalt and RavenCFD turbulence models perform comparable to NASAʼs verified models and since rigorous code verification is not possible without access to source code, the presented evidence suggests these turbulence models are implemented correctly for these or similar flow conditions and configurations. For solution verification, estimates of numerical uncertainty are less than 0.5% for 94% of the cases and less than 0.1% for 61% of the cases. For validation, the turbulent flat plate solutions match experiment skin friction within 4.8% for x/L>0.05, and for airfoil drag coefficient, S-A and SST agree within 1.2% of experiment, SARC 2%, and k-ω 4%.

Symmetric airfoil geometry effects on leading edge noise

Computational aeroacoustic methods are applied to the modeling of noise due to interactions between gusts and the leading edge of real symmetric airfoils. Single frequency harmonic gusts are interacted with various airfoil geometries at zero angle of attack. The effects of airfoil thickness and leading edge radius on noise are investigated systematically and independently for the first time, at higher frequencies than previously used in computational methods. Increases in both leading edge radius and thickness are found to reduce the predicted noise. This noise reduction effect becomes greater with increasing frequency and Mach number. The dominant noise reduction mechanism for airfoils with real geometry is found to be related to the leading edge stagnation region. It is shown that accurate leading edge noise predictions can be made when assuming an inviscid meanflow, but that it is not valid to assume a uniform meanflow. Analytic flat plate predictions are found to over-predict the noise due to a NACA 0002 airfoil by up to 3 dB at high frequencies. The accuracy of analytic flat plate solutions can be expected to decrease with increasing airfoil thickness, leading edge radius, gust frequency, and Mach number.

Evaluation of pipe bending reference stress equations

The use of a Failure Assessment Diagram (FAD) is widespread in the assessment of welddefects. To determine whether a defect is acceptable or not, this requires the calculation of a load ratio anda fracture ratio for the defect under consideration. Nowadays, many formulae are available to calculatethese two quantities and no clear guidance is given on which equation(s) should (not) be used. A partialclarification of this problem is achieved by comparing different reference stress equations. This article isconcerned with such comparison, for the specific case of welded pipes subjected to a bending load. A largeset of historical experimental data has been investigated in which defected pipes were subjected to anincreasing bending force until failure occurred. Two kinds of reference stress equations are considered, fullpipes subjected to a bending load and flat plates subjected to a uniform tension load. From the equationsunder consideration, the flat plate solution of Goodall & Webster and the empirical full pipe equations ofWilloughby and Wilkowski & Eiber describe the pipe failure in the most accurate way

Subsonic compressible flow in two-sided lid-driven cavity. Part I: Equal walls temperatures

This paper presents a numerical study of the laminar, viscous, subsonic compressible flow in a two-dimensional, two-sided, lid-driven cavity using a multi-domain spectral element method. The flow is driven by steadily moving two opposite walls vertically in opposite directions. All the bounding walls have equal temperatures. The results of the simulations are used to investigate the effects of the cavity aspect ratio, the Reynolds number and the Mach number on the flow. At lower Reynolds numbers, the flow pattern consists of two separate co-rotating vortices contiguous to the moving walls. For higher Reynolds numbers, initially a two-vortex flow is formed, which eventually turns into a single elliptical vortex occupying most of the cavity. For a higher aspect ratio, the flow patterns are dissimilar in that the streamlines become more and more elliptic. For aspect ratios as high as 2.5, at high Reynolds numbers, a three-vortex stage is formed. It is found that the compressibility effects are not very significant for Mach numbers less than 0.4. Dissipation of kinetic energy into internal energy changes the temperature field especially near the boundaries. Boundary layer studies suggest that the velocity and temperature boundary layer thicknesses are lower for higher Reynolds numbers. For engineering purposes, these thicknesses can be approximated by the existing flat-plate solutions.

Fluid flow and heat transfer in transitional boundary layers: effects of surface curvature and free stream velocity

Velocity and wall temperature measurements, over flat plate, concave and convex walls, were experimentally investigated in a low-speed wind tunnel with inlet velocities of 4 and 12 m/s encompassing the transitional region with streamwise distance Reynolds numbers from 3.15×105 to 1.04×106. As the velocity profiles, recorded by a semi-circular pitot tube and a digital constant-temperature hot-wire anemometer, were compared to exact Blasius profile and (1/7)th power law, experimental local Stanton numbers to analytical flat plate solution and turbulent correlation formula. Intermittency factors, derived from velocities and local Stanton numbers, were presented both in streamwise and pitchwise directions. It was found that the convex curvature delayed transition up to Re x =1.04×106, with a mean intermittency value of 0.61 and a shape factor of 1.81, where the similar intermittency and shape factors were determined at Re x of 8.33×105 and 4.25×105 for the flat plate and concave wall, indicating the enhancing role of concave curvature on the transition mechanism. The thinner boundary layers of the concave surface resulted in higher intermittency values, corresponding to higher skin friction and Stanton numbers; moreover the lowest gap between the measured and derived Stanton numbers were also obtained over the concave surface. Destabilising role of the concave wall caused Stanton numbers to increase up to 22%, whereas the convex wall, due to its stabilising character, produced lower Stanton numbers by 12% with respect to those of the flat plate.

Ground Testing Facility for Modeling Real Projectile Flight Heating in Earth Atmosphere

An analytical method for obtaining the heating of a hypersonic projectile with a conical forebody and data obtained from a special ballistic range to validate the method are presented. The method is a classical boundary-layer technique for turbulent flow. It involves a coordinate transformation that enables flat plate solutions to be adapted to the conical coordinates. The heating to the projectile nose cone was then obtained analytically with a semi-infinite slab, according to the given boundary conditions for flow velocity and gas temperature. The use of a transformation technique enables scaling of high-pressure conditions in the ballistic range, the so-called aerothermoballistic high-pressure test range, to actual atmospheric flight conditions. This facility allows duplication of the heating scenario during some seconds of atmosphere flight, with a flight in compressed gas resulting in the same heat flux for a flight time of some milliseconds. The transformation was validated using data from the range, obtained from the infrared surface emission at two wavelengths. Herewith the cone surface temperature is measured by detecting the intensity of the infrared emission coming from the cone surface

End-Capped Associative Polymer Chains between Nanospheres: Attractions in Ideal Solutions

We calculate equilibrium configurations of telechelic associative polymer chains without excluded volume interactions between flat plates and curved surfaces, obtaining analytical expressions for flat plates and numerical results for chains between two spheres. Ideal chains with strongly but reversibly adsorbed terminal groups produce an attraction of O(kT) per chain for planar surfaces as opposed to ≪ kT per chain in marginal and good solvents at high densities. Our numerical results for two spheres yield a weaker repulsion and an attraction of shorter range than predicted by the Derjaguin approximation based on the flat plate solution. These effects are due to deflection of chains out of the gap at close separations.

A new method for predicting stress intensity factors in cracked welded tubular joints

Fracture mechanics is an important tool for the analysis of cracked bodies. It has been established for use in a range of industries including the power generation and the aircraft industry. Fracture mechanics provides the basis for fatigue life prediction, steel selection and tolerance setting on allowable weld imperfections. It can also be used during the operational stage of a structure to make important decisions on inspection scheduling and repair strategies and as a tool for establishing limits on operational conditions. The stress intensity factor is a very important fracture mechanics parameter. Linear elastic fracture mechanics relies on the use of the stress intensity factor concept. Therefore the accuracy of any fracture mechanics model for the prediction of fatigue crack growth will depend very much on the accuracy of the stress intensity factor solution used. It is known that some factors are important in this process. One of these is the crack aspect ratio, which represents a major source of uncertainty in fatigue crack growth prediction. This paper presents a new approach for the prediction of stress intensity Y correction factors for welded tubular joints. The proposed method accounts for crack aspect ratio evolution during crack propagation. The method is based on a statistical model used to quantify and account for the deviation of experimental results from previous semi-empirical solutions and the modified Newman and Raju flat plate solution.

Numerical Study of Natural Convection in Fully Open Tilted Cavities

A numerical simulation of two-dimensional laminar natural convection in a fully open tilted square cavity with an isothermally heated back wall is conducted. The remaining two walls of the cavity are adiabatic. Steady-state solutions are presented for Grashof numbers between 102 and 105 and for tilt angles ranging from 60 to 90 (where 90 represents a cavity with the opening facing down). The fluid properties are assumed to be constant except for the density variation with temperature that gives rise to the buoyancy forces, which is treated by the Boussinesq approximation. The fluid concerned is air with Prandtl number fixed at 0.71. The governing equations are expressed in a normalized primitive variables formulation. Numerical predictions of the velocity and temperature fields are obtained using the finite-volume-based power law (SIMPLER: Semi-Implicit Method for Pressure-Linked Equations Revised) algorithm. For a vertical open cavity (alpha = 0 deg), the algorithm generated results that were in good agreement with those previously published. Flow patterns and isotherms are shown in order to give a better understanding of the heat transfer and flow mechanisms inside the cavity. Effects of the controlling parameters-Grashof number and tilt angle-on the heat transfer (average Nusselt number) are presented and analyzed. The results also revealed that the open-cavity Nusselt number approaches the flat-plate solution when either Grashof number or tilt angle increases. In addition, a correlation of the Nusselt number in terms of the Grashof number and tilt angle is developed and presented; a comparison is made with available data from other literature.

Bending of spherical lithosphere-axisymmetric case

SUMMARY Effects of sphericity are commonly ignored in the lithospheric bending problem. In order to examine its effects, I solve a simple axisymmetric spherical-shell model. The full solution and the asymptotic solution are derived from the basic equations, and their relationship to the flat-plate solution is examined. For displacement, effects of sphericity are small, and use of the flat-plate solution produces results that are numerically indistinguishable from those of the spherical solution. The most significant effect of sphericity appears in the stress, in particular the normal stress along the strike direction of the trench. This stress is approximately given by Eu,/R, where E is Young’s modulus, u, is the vertical deformation of the shell and R is its radius of curvature. If the shell (lithosphere) is bent downwards and reaches 30 km, this stress can become about 5 kbar in the Earth. While plastic behaviour may set in under such high pressure conditions and analysis beyond elasticity theory may be required, sphericity may be a cause of large compressive stress in the trench strike direction. This stress may play an important role in forming the overall shape of the Earth’s subduction zones.

Fluid Flow Phenomena in a Cylindrical Bath Agitated by Top Lance Gas Injection.

Fluid flow phenomena in a cylindrical bath agitated by gas injection from a submerged top lance were investigated on the basis of cold model experiments. A swirl motion of liquid, which was sumilar to the rotary sloshing, was observed under a certain blowing condition. The critical condition for the cessation of the swirl motion was first experimentally clarified and subsequently the following experiments were made in the absence of the swirl motion. The vertical migration distance of bubbles from the lance exit was measured using an electro-resistivity probe, and an empirical correlation for it was proposed as a function of the modified Froude number. The axial and radial velocity components of liquid were measured for a water-air system using a two-dimensional laser Doppler velocimeter. Applicability of previous emperical correlations of the axial mean velocity proposed for the gas injection through a centric bottom bozzle to the present case was examined. An electro-chemical method was used to measure the mass transfer coefficient between a platinum flat plate and aqueous FeSO4 solution. In the region below the bubble plume, the so-called dead water region was observed.

上吹き浸漬ランスからのガス吹込みを伴う円筒浴内の流動現象

Fluid flow phenomena in a cylindrical bath agitated by gas injection from a submerged top lance were investigated on the basis of cold model experiments. A swirl motion of liquid, which was similar to rotary sloshing, was observed under a certain blowing condition. The condition was experimentally clarified and subsequently the following experiments were made for cases without swirl motion. The vertical migration distance of gas from the lance exit was measured using an electro-resistivity probe and an empirical correlation of the migration distance was proposed as a function of the modified Froude number. The axial and radial velocity components of liquid phase were measured for a water-air system using a two-dimensional Laser Doppler Velocimeter. Applicability of the previous correlations of axial mean velocity proposed for gas injection through a centric bottom nozzle to the present case was examined. An electrochemical method was used to measure the mass transfer coefficient between a platinum flat plate and aqueous FeSO4 solution. In the axial region below the bubble plume, the so-called dead water region was observed.