Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries

In most practical engineering applications,the translating belt wraps around two fixed wheels.The boundary conditions of the dynamic model are typically specified as simply supported or fixed boundaries.In this paper,non-homogeneous boundaries are introduced by the support wheels.Utilizing the translating belt as the mechanical prototype,the vibration characteristics of translating Timoshenko beam models with nonhomogeneous boundaries are investigated for the first time.The governing equations of Timoshenko beam are deduced by employing the generalized Hamilton's principle.The effects of parameters such as the radius of wheel and the length of belt on vibration characteristics including the equilibrium deformations,critical velocities,natural frequencies,and modes,are numerically calculated and analyzed.The numerical results indicate that the beam experiences deformation characterized by varying curvatures near the wheels.The radii of the wheels play a pivotal role in determining the change in trend of the relative difference between two beam models.Comparing the results unearths that the relative difference in equilibrium deformations between the two beam models is more pronounced with smaller-sized wheels.When the two wheels are of equal size,the critical velocities of both beam models reach their respective minima.In addition,the relative difference in natural frequencies between the two beam models exhibits nonlinear variation and can easily exceed 50%.Furthermore,as the axial velocities increase,the impact of non-homogeneous boundaries on modal shape of translating beam becomes more significant.Although dealing with non-homogeneous boundaries is challenging,beam models with non-homogeneous boundaries are more sensitive to parameters,and the differences between the two types of beams undergo some interesting variations under the influence of non-homogeneous boundaries.

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

As a typical nonlinear wave,forward-leaning waves can be frequently encountered in the near-shore areas,which can impact coastal sediment transport significantly.Hence,it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately,especially for the turbulent boundary layer.In this work,the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied,and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer.Then,a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function.Consequently,an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation.The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature.

A Laboratory Modeling of the Velocity Field in the Convective Boundary Layer with the Particle Image Velocimetry Technique

Based on the research of the convective boundary layer (CBL) temperature field in a convective tank, this paper studies the characteristics of the CBL velocity field in the convective tank. Aluminium powder (400 orders) is used as a tracer particle in the application of the particle image velocimetry (PIV) technique. The experiment demonstrates: the velocity distribution in the mixed layer clearly possesses the characteristics of CBL thermals; the velocity distribution in the top zone of the mixed layer shows entrainment layer characteristics; the vertical distribution of turbulent characteristic variables is reasonable, which is similar to field observations and other tank results; the error analysis demonstrates the validity of aluminium powder, which implies the reliability of the results.

Simulation of the seismic response of sedimentary basins with constant-gradient velocity along arbitrary direction using boundary element method:SH case

We presented a boundary element method using the approximate analytical Green＇s function given by Sanchez-Sesma et al. Coordinate transform is introduced to extend the method to deal with the model with constant-gradient velocity along oblique direction. The method is validated by comparing the numerical results with other independent methods. This method provides a useful tool for analyzing local site effects. We computed seismic response for two series of models. The results in both frequency and time domains are analyzed and show complex amplification patterns. The fundamental mode of resonance is dependent not only on the velocity at the free surface but also on the velocity distribution of the whole basin. For the higher modes of vibration the heterogeneous basin also has its own characteristic.

A Laboratory Study of the Turbulent Velocity Characteristics in the Convective Boundary Layer

Based on the measurement of the velocity field in the convective boundary layer （CBL） in a convection water tank with the particle image velocimetry （PIV） technique, this paper studies the characteristics of the CBL turbulent velocity in a modified convection tank. The experiment results show that the velocity distribution in the mixed layer clearly possesses the characteristics of the CBL thermals, and the turbulent eddies can be seen obviously. The comparison of the vertical distribution of the turbulent velocity variables indicates that the modeling in the new tank is better than in the old one. The experiment data show that the thermal＇s motion in the entrainment zone sometimes fluctuates obviously due to the intermittence of turbulence. Analyses show that this fluctuation can influence the agreement of the measurement data with the parameterization scheme, in which the convective Richardson number is used to characterize the entrainment zone depth. The normalized square velocity wi^2/w＊^2. at the top of the mixed layer seems to be time-dependent, and has a decreasing trend during the experiments. This implies that the vertical turbulent velocity at the top of the mixed layer may not be proportional to the convective velocity （w＊）.

Experimental studies on flow visualization and velocity field of compression ramp with different incoming boundary layers

Experimental studies which focus on flow visualization and the velocity field of a supersonic laminar/turbulent flow over a compression ramp were carried out in a Mach 3.0 wind tunnel. Fine flow structures and velocity field structures were obtained via NPLS （nanoparticle-tracer planar laser scattering） and PIV （particle image velocimetry） techniques, time- averaged flow structures were researched, and spatiotemporal evolutions of transient flow structures were analyzed. The flow visualization results indicated that when the ramp angles were 25~, a typical separation occurred in the laminar flow, some typical flow structures such as shock induced by the boundary layer, separation shock, reversed flow and reattachment shock were visible clearly. While a certain extent separation occurred in turbulent flow, the separation region was much smaller. When the ramp angles were 28~, laminar flow separated further, and the separation region expanded evidently, flow structures in the separation region were complex. While a typical separation occurred in turbulent flow, reversed flow structures were significant, flow structures in the separation region were relatively simple. The experimental results of velocity field were corresponding to flow visualization, and the velocity field structures of both compression ramp flows agreed with the flow structures well. There were three layered structures in the U component velocity, and the V component velocity appeared like an oblique ＂v＂. Some differences between these two compression ramp flows can be observed in the velocity profiles of the shear layer and the shearing intensity.

Error Analysis of A New Higher Order Boundary Element Method for A Uniform Flow Passing Cylinders

A higher order boundary element method(HOBEM)is presented for inviscid flow passing cylinders in bounded or unbounded domain.The traditional boundary integral equation is established with respect to the velocity potential and its normal derivative.In present work,a new integral equation is derived for the tangential velocity.The boundary is discretized into higher order elements to ensure the continuity of slope at the element nodes.The velocity potential is also expanded with higher order shape functions,in which the unknown coefficients involve the tangential velocity.The expansion then ensures the continuities of the velocity and the slope of the boundary at element nodes.Through extensive comparison of the results for the analytical solution of cylinders,it is shown that the present HOBEM is much more accurate than the conventional BEM.

Prediction of boundary shear stress distribution in straight open channels using velocity distribution

Conventional methods for measuring local shear stress on the wetted perimeter of open channels are related to the measurement of the very low velocity close to the boundary.Measuring near-zero velocity values with high fluctuations has always been a difficult task for fluid flow near solid boundaries.To solve the observation problems,a new model was developed to estimate the distribution of boundary shear stress from the velocity distribution in open channels with different cross-sectional shapes.To estimate the shear stress at a point on the wetted perimeter by the model,the velocity must be measured at a point with a known normal distance to the boundary.The experimental work of some other researchers on channels with various cross-sectional shapes,including rectangular,trapezoidal,partially full circular,and compound shapes,was used to evaluate the performance of the proposed model.Optimized exponent coefficients for the model were found using the multivariate Newton method with the minimum of the mean absolute percentage error(MAPE)between the model and experimental data as the objective function.Subsequently,the calculated shear stress distributions along the wetted perimeter were compared with the experimental data.The most important advantage of the proposed model is its inherent simplicity.The mean MAPE value for the seven selected cross-sections was 6.9%.The best results were found in the cross-sections with less discontinuity of the wetted perimeter,including the compound,trapezoidal,and partially full circular pipes.In contrast,for the rectangular cross-section with an angle between the bed and walls of 90°,MAPE increased due to the large discontinuities.

An approach to determine coeffi cients of logarithmic velocity vertical profile in the bottom boundary layer

Velocity vertical profiles in the bottom boundary layer are important to understand the oceanic circulation.The logarithmic vertical profile,u=A ln z+B,is the universal profile for the horizontal velocity in the boundary layer,in which two coefficients(A and B)need to be determined.The two coefficients are the functions of the friction velocity(u_(*))and the roughness length(z_(0)),and they are calculated using u_(*)and z_(0).However,the measurement of u_(*)and z_(0) is a challenge.In the present study,an approach is developed to estimate the two coefficients(A and B)by using a series of fl ume laboratory experiments with fl at boundary and regularly distributed cylinders as the rough boundaries.An acoustic doppler velocimeter(ADV)is used to measure the velocity vertical profiles of the steady flow.Using the measured velocity data,the regressed logarithmic profiles are obtained.Based on the series of the A and B values,the mathematical formula for A and B are statistically established as the function of the cylinder height,inflow velocity,and the water depth,which avoids the measurement of the friction velocity and the roughness length.

PRELIMINARY STUDY ON THE FORMATION MECHANISM OF COUNTER WESTERN BOUNDARY UNDERCURRENTS BELOW THE THERMOCLINE——A CONCEPTUAL MODEL

Based on a simple conceptual model of stratified ocean, the criterion of the geostrophic velocity inversion in and below the thermocline was derived as h ′·η′<0 and ρ 1|η′|≤Δρ| h ′|, meaning that the slopes of the thermocline( h ′) and the sea surface(η′) must be opposite to each other, and that h ′ must be strong enough to satisfy the latter inequality. The criterion was applied to discuss the features of the western boundary undercurrents, the counter undercurrents of the western boundary currents below the thermocline, and to discuss the dynamics of their formation finally resulting from the combination of the basin scale circulation and local geostrophic balance. The formation mechanism, multi core structure, and transport variations of the Mindanao Undercurrent and those of other undercurrents, such as the North Equatorial Undercurrent and the Kuroshio undercurrent, can be satisfactorily explained by the above results.

EFFECT OF THE RIBLETS SURFACE ON THE BOUNDARY LAYER DEVELOPMENT

The effect of riblets surface on flat plate boundary layer development hasbeen measured with laser Doppler velocimeter (LDV). In the present experiment, although the development of boundary layer can be divided into three regions, the laminarboundary layer region is increased, and the transition occurs at a larger Reynolds number.The Reynolds number of transition to turbulence Rextr is 4. 3 tinies as great as that in thesmooth flat plate case. In the turbulent boundary layer region, greater value of integralconstant C is obtained in log-law for velocity distribution. This indicates that the ribletssurface has obviously drag reduction characteristics. The maximum value of turbulence intensity and its position are less than the smooth surface results in the turbulent boundarylayer region.

A Kinematic Thermal Model for Descending Slabs with Velocity Boundary Layers:A Case Study for the Tonga Subducting Slab

For the purpose of investigating the influence of metastable olivine（MO） phase transformations on both deep seismicity and stagnation of slabs,we constructed a 2-dimensional finite element thermal model for a 120 Ma-old 50°dipping oceanic lithosphere descending at 10 cm/yr with velocity boundary layers,which would mitigate the interference of constant velocity field for the slab. The resulting temperatures show that most of intermediate and deep earthquakes occurring within the Tonga slab are occurring inside the 800℃and 1200℃isotherm,respectively.The elevation of olivine transformation near～410 km and respective persistence of metastable olivine and spinel within the transition zone and beneath 660 km would thus result in bimodal positive,zonal,negative density anomalies,respectively.These results together with the resulting pressure anomalies may reflect the stress pattern of the Tonga slab：（i） slab pull force exerts above a depth of～230 km;（ii） MO existence changes the buoyancy force within the transition zone and facilitates slab stagnation at a depth of 660 km;（iii） as the subducting materials accumulated over 660 km,deepest earthquakes occur due to MO transformation;（iv） a flattened‘slab’ may penetrate into the lower mantle due to the density increment of Sp transformation.

Numerical Simulation for Water Entry of a Wedge at Varying Speed by a High Order Boundary Element Method

A high order boundary element method was developed for the complex velocity potential problem. The method ensures not only the continuity of the potential at the nodes of each element but also the velocity. It can be applied to a variety of velocity potential problems. The present paper, however, focused on its application to the problem of water entry of a wedge with varying speed. The continuity of the velocity achieved herein is particularly important for this kind of nonlinear free surface flow problem, because when the time stepping method is used, the free surface is updated through the velocity obtained at each node and the accuracy of the velocity is therefore crucial. Calculation was made for a case when the distance S that the wedge has travelled and time t follow the relationship s=Dtα, where D and α are constants, which is found to lead to a self similar flow field when the effect due to gravity is ignored.

Influence of Rigid Boundary and Initial Stress on the Propagation of Love Wave

In the present paper we study the effect of rigid boundary on the propagation of Love waves in an inhomogeneous substratum over an initially stressed half space, where the heterogeneity is both in rigidity and density. The dispersion equation of the phase velocity has been derived. It has been found that the phase velocity of Love wave is considerably influenced by the rigid boundary, inhomogeneity and the initial stress present in the half space. The velocity of Love waves have been calculated numerically as a function of KH (where K is a wave number H is a thickness of the layer) and are presented in a number of graphs.

Characteristic Study of the Boundary Layer Parameters over the Arabian Sea and the Bay of Bengal Using the QuikSCAT Dataset

The marine atmospheric boundary layer （MABL） plays a vital role in the transport of momentum and heat from the surface of the ocean into the atmosphere. A detailed study on the MABL characteristics was carried out using high-resolution surface-wind data as measured by the QuikSCAT （Quick scatterometer） satellite. Spatial variations in the surface wind, frictional velocity, roughness parameter and drag coefficient for the different seasons were studied. The surface wind was strong during the southwest monsoon season due to the modulation induced by the Low Level Jetstream. The drag coefficient was larger during this season, due to the strong winds and was lower during the winter months. The spatial variations in the frictional velocity over the seas was small during the post-monsoon season （-0.2 m s^-1）. The maximum spatial variation in the frictional velocity was found over the south Arabian Sea （0.3 to 0.5 m s^-1） during the southwest monsoon period, followed by the pre-monsoon over the Bay of Bengal （0.1 to 0.25 m s^-1）. The mean wind-stress curl during the winter was positive over the equatorial region, with a maximum value of 1.5×10^-7 N m^-3, but on either side of the equatorial belt, a negative wind-stress curl dominated. The area average of the frictional velocity and drag coefficient over the Arabian Sea and Bay of Bengal were also studied. The values of frictional velocity shows a variability that is similar to the intraseasonal oscillation （ISO） and this was confirmed via wavelet analysis. In the case of the drag coefficient, the prominent oscillations were ISO and quasi-biweekly mode （QBM）. The interrelationship between the drag coefficient and the frictional velocity with wind speed in both the Arabian Sea and the Bay of Bengal was also studied.

MULTI-SCALE COHERENT STRUCTURES IN TURBULENT BOUNDARY LAYER DETECTED BY LOCALLY AVERAGED VELOCITY STRUCTURE FUNCTIONS

The time sequence of longitudinal velocity component at different vertical locations in turbulent boundary layer was finely measured in a wind tunnel. The concept of coarse_grained velocity structure functions, which describes the relative motions of straining and compressing for multi_scale eddy structures in turbulent flows, was put forward based on the theory of locally multi_scale average. Based on the consistency between coarse_grained velocity structure function and Harr wavelet transformation,detecting method was presented, by which the coherent structures and their intermittency was identified by multi_scale flatness factor calculated by locally average structure function. Phase_averaged evolution course for multi_scale coherent eddy structures in wall turbulence were extracted by this conditional sampling to educe scheme. The dynamics course of multi_scale coherent eddy structures and their effects on statistics of turbulent flows were studied.

EFFECTS OF TURBULENT COHERENT STRUCTURE ON PLUME DIFFUSION IN ATMOSPHERE BOUNDARY LAYER

The simulation tests of plume diffusion have been carried out in an open water channel,in which a Laser Doppler Velocitymeter,optical concentration measurement and flow visualization technology have been used.Through experiments and theoretical analyses,the effects of environmental wind field,surface roughness and discharge condition of elevated source on plume diffusion have been investigated.The plume dif- fusion mechanism has been elucidated reasonably,based on the turbulent coherent structure theory.

A Kinematic Thermal Model for Descending Slabs with Velocity Boundary Layers

Kinematic models compute the temperature distribution by prescribing a constant convergent velocity for the subducting slab,resulting in an artificial velocity discontinuity,which may accelerate the heating of subducting slabs.For the purpose to moderate the influence of such artificial discontinuity, we construct a 2D thermal model for subduction zones with a velocity boundary layer,within which the velocities decrease linearly with the distance from the interfaces of slabs.Temperatures are calculated

Investigation of Wall Pressure Fluctuations in a Turbulent Boundary Layer by Large Eddy Simulation

Large eddy simulation (LES) was used to investigate the space-time field of the low Mach number, fully developed turbulent boundary layer on a smooth, rigid flat plate. The wall-pressure field simulated by LES was analyzed to obtain the pressure statistics, including the wall-pressure root-mean square, skewness and flatness factors, which show the wall pressure distribution was not Gaussian. The profile of the auto-power spectral density and the contour of the streamwise wavenumber-frequency spectral density of wall-pressure were plotted. The "convection ridge" can be observed clearly and the convection velocity can be calculated from the location of the convection peak.

Effects of Orbital Motion on the Boundary Layer Flow on a Spinning Disk

The objective of this study is to experimentally examine the effects of orbital motion on the boundary layer flow on a spinning disk. The boundary layer flow on the disk is visualized by the oil flow method, and velocity in the boundary layer is measured by the hot-wire method. For the oil flow pattern in the case of spinning motion only, streaks are clearly observed on the disk as transient vortices, but by adding orbital motion to the spinning motion, we find that streaks are not observed in a certain range of orbital conditions. With increasing orbital motion speed, the laminar re- gion becomes narrower and transition is promoted from the inward region of the disk, regardless of the direction of ro- tation. Also, with the addition of orbital motion, the velocity profile in the boundary layer becomes more asymmetric with respect to the spin axis of the disk. Furthermore, stationary vortices do not appear on the disk when the orbital speed is beyond a certain critical value. Therefore, the lack of streaks in the oil film pattern when orbital motion is added is due to the spatiotemporal unsteadiness of the flow field on the disk.