The stress state of a plate subjected to a piecewise homogeneous load

We consider the stress-strain state of a plate having a doubly connected domain S bounded from the outside by a circle of radius R and from the inside by an ellipse with two rectilinear cuts. The cuts lie symmetrically on the x-axis. The plate is subjected to various forces: the hole contour (the ellipse) is under the action of uniformly distributed forces of intensity q, and the cut shores are free of loads; at the points ±ib of the imaginary axis, the plate is under the action of a lumped force P.The solution of the problem is reduced to determining two analytic functions φ(z) and ψ(z) satisfying certain boundary conditions (depending on the type of the acting loads).We use the Kolosov-Muskhelishvili method to reduce the problem to a system of linear algebraic equations for the coefficients in the expansions of the functions φ(z) and ψ(z). The solution thus obtained is illustrated by numerical examples.     

Non-symmetric bi-stable flow around the Ahmed body

The flow around the Ahmed body at varying Reynolds numbers under yawing conditions is investigated experimentally. The body geometry belongs to a regime subject to spanwise flow instability identified in symmetric flow by Cadot and co-workers (Grandemange et al., 2013b). Our experiments cover the two slant angles 25° and 35° and Reynolds numbers up to 2.784 × 10⁶. Special emphasis lies on the aerodynamics under side wind influence. For the 35° slant angle, forces and moments change significantly with the yawing angle in the range 10° ≤ |β| ≤ 15°. The lift and the pitching moment exhibit strong fluctuations due to bi-stable flow around a critical angle β of ±12.5°, where the pitching moment changes sign. Time series of the forces and moments are studied and explained by PIV measurements in the flow field near the rear of the body.     

Vibrations of a polygonal plate weakened by a circular hole with two rectilinear cuts

We consider free bending vibrations of a finite isotropic plate bounded on the outside by a polygon L ₂ (e.g., a square, an ellipse, a circle, etc.) and on the inside by the contour L ₁ formed by the circle of radius r with two rectilinear cuts located symmetrically on the axis Ox. The plate is rigidly clamped along the entire outer contour L ₂, and the inner contour L ₁ is free. We reduce the solution of the plate vibration problem to the integration of a fourth-order differential equation [2, 4, 5]. The method used in this paper is well known in the literature [2, 4, 5, 11] for simple simply connected domains, but the case under study (a doubly connected domain with cuts) has not yet been considered, because no mapping functions z = λ(ξ) have been known for complicated domains (doubly connected domain with cuts). The author [6–8] is the first in the scientific world to find such mapping functions. The obtained theoretic solution is illustrated by numerical examples.     

Flow around two tandem square cylinders near a plane wall

An experimental study has been conducted to investigate the flow around two identical square cylinders in tandem arrangement and placed near a plane wall at a Reynolds number of 6,300. The inter-cylinder spacing ratio was varied from S ^* = 0.5 to 6, and the cylinder-to-wall gap ratio from G ^* = 0.25 to 2. Totally, 42 cases were considered to systematically examine the effects of wall proximity and the mutual interference between the two cylinders in the normalized gap–spacing (G ^*–S ^*) plane. The flow fields were captured using digital particle image velocimetry, in conjunction with measurements of the fluid forces (drag and lift) acting on the downstream cylinder using a piezoelectric load cell. The results show that the flow is highly dependent on the combined values of G ^* and S ^*. Categories relating to G ^* could be broadly classified as small-gap regime (G ^* < 0.5) at which periodic vortex shedding from the cylinders is suppressed, intermediate-gap regime (0.5 < G ^* < 1) where vortex shedding occurs but is under the influence of the wall proximity, and large-gap regime (G ^* > 1) where the wall effects become negligible. Similarly, the flow interference between the two cylinders can be divided into three basic categories as a function of S ^*, namely, shielding regime at S ^* < 1, reattachment regime at 1 < S ^* < 3, and impinging regime at S ^* > 3. Variations of force coefficients, amplitude spectra, Strouhal numbers, and Reynolds shear stress with G ^* and S ^* are presented to characterize the different flow regimes.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139     

Emission of organic carbon,elemental carbon and water-soluble ions from crop straw burning under flaming and smoldering conditions

Emissions from major agricultural residues were measured using a self-designed combustion system. Emission factors (EFs) of organic carbon (OC), elemental carbon (EC), and water-soluble ions (WSIs) (K⁺, NH₄⁺, Na⁺, Mg²⁺, Ca²⁺, Cl⁻, NO₃⁻, SO₄^2–) in smoke from wheat and rice straw were measured under flaming and smoldering conditions. The OC₁/TC (total carbon) was highest (45.8% flaming, 57.7% smoldering) among carbon fractions. The mean EFs for OC (EF_OC) and EC (EF_EC) were 9.2 ± 3.9 and 2.2 ± 0.7 g/kg for wheat straw and 6.4 ± 1.9 and 1.1 ± 0.3 g/kg for rice straw under flaming conditions, while they were 40.8 ± 5.6 and 5.8 ± 1.0 g/kg and 37.6 ± 6.3 and 5.0 ± 1.4 g/kg under smoldering conditions, respectively. Higher EC ratios were observed in particulate matter (PM) mass under flaming conditions. The OC and EC for the two combustion patterns were significantly correlated (p < 0.01, R = 0.95 for wheat straw; p < 0.01, R = 0.97 for rice straw), and a higher positive correlation between OC₃ and EC was observed under both combustion conditions. WSIs emitted from flaming smoke were dominated by Cl⁻ and K⁺, which contributed 3.4% and 2.4% of the PM mass for rice straw and 2.2% and 1.0% for wheat straw, respectively. The EFs of Cl⁻ and K⁺ were 0.73 ± 0.16 and 0.51 ± 0.14 g/kg for wheat straw and 0.25 ± 0.15 and 0.12 ± 0.05 g/kg for rice straw under flaming conditions, while they were 0.42 ± 0.28 and 0.12 ± 0.06 g/kg and 0.30 ± 0.27 and 0.05 ± 0.03 g/kg under smoldering conditions, respectively. Na⁺, Mg²⁺, and NH₄⁺ were vital components in PM, comprising from 0.8% (smoldering) to 3.1% (flaming) of the mass. Strong correlations of Cl⁻ with K⁺, NH₄⁺, and Na⁺ ions were observed in rice straw and the calculated diagnostic ratios of OC/EC, K⁺/Na⁺ and Cl⁻/Na⁺ could be useful to distinguishing crop straw burning from other sources of atmospheric pollution.     

Viscous flow phenomena in a partially filled rotor–stator system

This paper deals with free-surface flows in a geometrically simple apparatus consisting of a rotating cylindrical drum with horizontal axis and a fixed internal plate. The drum is partially filled with a very viscous, optically transparent liquid and is otherwise filled with air as a passive phase. Due to the rotation some liquid mass is dragged upwards along the moving wall, scraped off by the fixed plate and forced to fall back into the liquid pool due to gravity. The resulting flow and the shape of the phase interface are visualised within a plane perpendicular to the cylinders axis. Different flow phenomena are detected, depending on the volumetric filling degree F and on the dimensionless parameter Λ  =  gR/(ν Ω), which quantifies the relative influence of gravitational and viscous forces. Extended lubrication theories are used in order to model the shape of the phase interphase and to predict the conditions under which inner film flows may be expected. Some findings are validated by results of a more sophisticated numerical free-surface-flow simulation.     

Experimental and numerical study of turbulent flow and heat transfer inside hexagonal duct

Flow and heat transfer characteristics in transition and turbulent regions are studied experimentally and numerically in a horizontal smooth regular hexagonal duct under constant wall temperature boundary condition covering a range of Reynolds number from 2.3 × 10³ to 52 × 10³. Two types of k-omega (standard and shear stress transport (SST)) and three types of k-ε (standard, renormalization (RNG), and realizable) turbulence model are employed for transition and turbulent regions, respectively. Both average and fully developed Darcy friction factor and Nusselt number are presented as a function of Reynolds number. It is seen that k-omega SST and k-ε realizable turbulence models gave the best agreement with the experimental data in transition and turbulent regions, respectively. All the experimental results are correlated within an accuracy of ±13 % and ±7 % for Nusselt number and Darcy friction factor, respectively. Results obtained in this study are compared with circular duct results using hydraulic diameter.     

Aerodynamics of laminar separation flutter at a transitional Reynolds number

Experimental observations of self-sustained pitch oscillations of a NACA 0012 airfoil at transitional Reynolds numbers were recently reported. The aeroelastic limit cycle oscillations, herein labelled as laminar separation flutter, occur in the range 5.0×10⁴≤Re_c≤1.3×10⁵. They are well behaved, have a small amplitude and oscillate about θ=0°. It has been speculated that laminar separation leading to the formation of a laminar separation bubble, occurring at these Reynolds numbers, plays an essential role in these oscillations. This paper focuses on the Re_c=7.7×10⁴ case, with the elastic axis located at 18.6% chord. Considering that the experimental rig acts as a dynamic balance, the aerodynamic moment is derived and is empirically modelled as a generalized Duffing–van-der-Pol nonlinearity. As expected, it behaves nonlinearly with pitch displacement and rate. It also indicates a dynamically unstable equilibrium point, i.e. negative aerodynamic damping. In addition, large eddy simulations of the flow around the airfoil undergoing prescribed simple harmonic motion, using the same amplitude and frequency as the aeroelastic oscillations, are performed. The comparison between the experiment and simulations is conclusive. Both approaches show that the work done by the airflow on the airfoil is positive and both have the same magnitude. The large eddy simulation (LES) computations indicate that at θ=0°, the pitching motion induces a lag in the separation point on both surfaces of the airfoil resulting in negative pitching moment when pitching down, and positive moment when pitching up, thus feeding the LCO.     

Determination of Bond Strengths in Non-woven Fabrics: a Combined Experimental and Computational Approach

Interfiber bonds are important structural components in non-woven fabrics. Bond fracture greatly affects the strength and damage progression in a fiber network structure. Here, we present a novel combined experimental and computational approach to extract bond strengths in non-wovens. In this method, a small specimen is imaged and the obtained 3D geometry of the network is directly modeled in a finite element framework. Bond properties are determined by matching finite element simulation predicted mechanical response to the experimental data. This method is demonstrated by applying it to six specimens of a commercial polypropylene non-woven. A four parameter bi-linear interface law is used with normal stiffness k, shear stiffness βk, separation at the start of damage d ₁, and separation at total loss of bond stiffness d ₂. The determined normal strength (kd ₁)and shear strength (βkd ₁) are (1.3 ± 0.3) × 10² MPa and (1.0 ± 0.2) × 10² MPa, respectively. To show that the obtained bond parameters can be applied to a new specimen, a cross validation is conducted whereby parameters are fit from five specimens and then evaluated on the sixth. Additional validation of the obtained bond strength parameters was conducted with larger size artificial network simulations and peel tests. The proposed method in this work carries the dual advantages of characterizing actual bonds in a non-woven and characterizing hundreds of bonds simultaneously. The method can be applied to a variety of non-woven fabrics that are bonded at fiber-fiber intersections.     

The effect of a small isolated roughness element on the forces on a sphere in uniform flow

The effect of an isolated roughness element on the forces on a sphere was examined for a Reynolds number range of 5 × 10⁴ < Re < 5 × 10⁵ using a novel sting-mounted sphere apparatus. The roughness element was a circular cylinder, and its width and height was varied to be 1, 2, and 4% of the sphere diameter. At subcritical Re, a lateral force is produced in the direction of the roughness, while at supercritical Re, the force is in the opposite direction. This is caused by asymmetric boundary layer separation, as shown using particle image velocimetry. At supercritical Re, a roughness element that is only 1% the sphere diameter produces a lift to drag ratio of almost one. It was found that the isolated roughness element has the largest effect on the lateral forces when it is located between a streamwise angle of about 40° and 80°. In addition to the mean forces, the unsteady forces were also measured. It was found that at subcritical Re, vortex shedding is aligned to the plane of the roughness element. In addition, the probability distribution of the forces was nearly Gaussian for subcritical Re, but for supercritical Re, the skewness and kurtosis deviate from Gaussian, and the details are dependent on the roughness size. A simple model developed for the vortical structure formed behind the roughness element can be extended to explain aspects of nominally smooth sphere flow, in which external disturbances perturb the sphere boundary layer in an azimuthally local sense. These results also form the basis of comparison for an investigation into the effectiveness of a moving isolated roughness element for manipulating sphere flow.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

The aerodynamics of a cylinder submerged in the wake of another

Flow-induced fluctuating lift (C_Lf) and drag (C_Df) forces and Strouhal numbers (St) of a cylinder submerged in the wake of another cylinder are investigated experimentally for Reynolds number (Re)=9.7×10³–6.5×10⁴. The spacing ratio L^⁎ (=L/D) between the cylinders is varied from 1.1 to 4.5, where L is the spacing between the cylinders and D is the cylinder diameter. The results show that C_Lf, C_Df and St are highly sensitive to Re due to change in the inherent nature of the flow structure. How the flow structure is dependent on Re and L^⁎ is presented in a flow structure map. Zdravkovich and Pridden (1977) observed a ‘kink’ in time-mean drag distribution at L^⁎≈2.5 for Re>3.1×10⁴, but not for Re≤3.1×10⁴. The physics is provided here behind the presence and absence of the ‘kink’ that was left unexplained since then.     

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

The reflection of a symmetric Rayleigh-Lamb wave at the fixed or free edge of a plate

A semi-infinite plate of homogeneous isotropic, linearly elastic material occupies the region x≥0, |y|≤1, -∞<z<∞; the faces y=±1 are free of tractions, the end x=0 may be either fixed or traction free, and there are no body forces. A plane strain, time-harmonic, symmetric Rayleigh-Lamb wave propagates in the plate and is normally incident upon the end x=0. The problem of determining the resulting reflected wave field is solved by the “method of projection”, a method developed by the authors for solving corresponding problems in elastostatics. The solutions obtained for the dynamic problem fully satisfy the equations and boundary conditions of the linear theory, and (in the fixed-end case) proper account is taken of the singularities of the stress field at the corners x=0, y=±1. In each case the division of energy between the various reflected modes is found, and the dynamical stress intensity factors at the corners are determined in the fixed-end case. The existence of an “edge-mode” for the free-end case at a single isolated value of the frequency is confirmed, but a careful search revealed no similar phenomenon for the fixed-end case.     

Flow characteristics and flow-induced forces of a stationary and rotating triangular cylinder with different incidence angles at low Reynolds numbers

In this paper, the problem of two-dimensional fluid flow past a stationary and rotationally oscillating equilateral triangular cylinder with a variable incident angle, Reynolds number, oscillating amplitude, and oscillating frequency is numerically investigated. The computations are carried out by using a two-step Taylor-characteristic-based Galerkin (TCBG) algorithm. For the stationary cases, simulations are conducted at various incident angles of α=0.0–60.0° and Reynolds numbers of Re=50–160. For the oscillation cases, the investigations are done at various oscillating amplitudes of θ_max=7.5–30.0° and oscillating frequencies of F_s/F_o=0.5–3.0 considering two different incidence angles (α=0.0°, 60.0°) and three different Reynolds numbers (Re=50, 100, 150). The results show that the influences of key parameters (incidence angle, Reynolds number, oscillating amplitude, and oscillating frequency) are significant on the flow pattern and hydrodynamic forces. For the stationary cases, at smaller angle of incidence (α≤30.0°), Reynolds number has a large impact on the position of the separation points. When α is between 30.0° and 60.0°, it was found that the separation points are located at the rear corners. From a topological point of view, the diagram of flow pattern is summarized, including two distinct patterns, namely, main separation and vortex merging. A deep analysis of the influence of Reynolds number and incidence angles on the mean pressure coefficient along the triangular cylinder surface is presented. Additionally, for the oscillating cases, the lock-on phenomenon is captured. The dominant flow patterns are 2S mode and P+S mode in lock-on region at α=0.0°. It is found at α=60.0°, however, that the flow pattern is predominantly 2S mode. Furthermore, except for the case of F_s/F_o=2.0, the mean drag decreases as the oscillating amplitude increases for each Reynolds number at α=0.0°. At α=60.0°, the minimum mean drag for F_s/F_o=1.5 is lower than that for stationary case, and occurs at θ_max=15.0° (Re=100) and θ_max=22.5° (Re=150), respectively. Finally, the effect of Reynolds number on a rotational oscillation cylinder is elucidated.     

Perforation of 6082-T651 Aluminum Plates with 7.62 mm APM2 Bullets at Normal and Oblique Impacts

We conducted an experimental study to understand the mechanisms and dominant parameters for 7.62 mm APM2 bullets that perforate 6082-T651 aluminum armor plates at oblique impacts. The 7.62-mm-diameter, 10.7 g, APM2 bullet consists of a brass jacket, lead filler, and a 5.25 g, ogive-nose, hard steel core. The brass and lead were stripped from the APM2 bullets by the targets, so we conducted ballistic experiments with both the APM2 bullets and only the hard steel cores. These projectiles were fired from a rifle to striking velocities between 400 and 1,000 m/s into 20-mm-thick plates at normal impact (β?=?0^o) and at oblique angles of β?=?15^o, 30^o, and 45^o. Measured residual and ballistic-limit velocities for the full bullet and the hard core were within a few percent for normal impact and all oblique angles. Thus, we showed that the perforation process was dominated by the hard steel core of the bullet. In addition, we conducted large strain, compression tests on the 6082-T651 plate material for input to perforation equations derived from a cavity-expansion model for the steel core projectiles. Model predictions were shown to be in good agreement with measured ballistic-limit and residual velocity measurements for β?=?0^o, 15^o, and 30^o. We also presented a scaling law for the bullet that showed the ballistic-limit velocities were proportional to the square root of the product of plate thickness and a material strength term.     

Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks

The propagation of time-harmonic waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x = md, y = lh(m, l = 0, ±1, ±2,…). The wave motion is polarized in the z-direction and propagates in the y-direction (normal to the cracks). The theory of Floquet or Bloch waves, together with an appropriate Green's function and the condition of vanishing traction on the crack faces leads to a system of singular integral equations, which provides the basis for the derivation of an exact dispersion equation. Numerical results are presented for the wave number as a function of the frequency. The frequency spectrum shows a pattern of passing and stopping bands. The exact results are compared with the frequency spectrum according to a simplified theory which considers the arrays of collinear cracks in the planes y = lh (l= 0 ±1, ±2,…) as planes of homogeneous transmission and reflection. Good agreement is observed between exact and approximate results.     

Combined forced and free convective flow of water at 4°C past a semi-infinite vertical plate

Effects of buoyancy forces on forced and free convective flow of water at 4°C past a semi-infinite vertical plate at constant temperature are studied. Flow is assumed to be vertically upwards. Similarity solutions are derived and the resulting equations are solved numerically on a computer. Velocity and temperature profiles are shown graphically and numerical values of the skin friction and the rate of heat transfer are entered in tables. It is observed that the skin friction and the Nusselt number increase with increasing Gr/Re², where Gr is the Grashof number and Re is the Reynolds number     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.