Separation zones in the flow near a small-angle convex corner

On the basis of an asymptotic analysis of the Navier-Stokes system of equations for large Reynolds numbers (Re → ∞), the plane incompressible fluid flow near a surface having a convex corner with a small angle 2θ* is investigated. It is shown that for θ* = O(Re^?1/4), in addition to the known solution that describes a separated flow completely localized in a thin “viscous” sublayer of the interaction region near the corner point, another solution corresponding to a flow with a developed separation zone is possible. For θ ₀ = Re^1/4 θ* = O(1), the longitudinal dimension of this zone varies from finite values up to values of the order of Re^?3/8. The nonuniqueness of the solution is established on a certain range of variation of the parameter θ ₀. The dependence of the drag coefficient on the angle θ* is found.     

Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading

In this paper, we compute the location of the imminent hinges and the minimum thickness, t, of a circular masonry arch with mid-thickness radius, R, and embracing angle, β, which can just sustain its own weight together with a given level of a horizontal ground acceleration, ε g. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures in earthquake prone areas, this paper shows that the value of the minimum horizontal acceleration that is needed to convert an arch with slenderness (t/R, β) into a four-hinge mechanism depends on the direction of the rupture at the imminent hinge locations. This result is obtained with a variational formulation and the application of the principle of stationary potential energy, and it is shown that a circular arch becomes a mechanism with vertical ruptures when subjected to a horizontal ground acceleration that is slightly lower than the horizontal acceleration needed to create a mechanism with radial ruptures. The paper explains that the multiplicity on the solution for the minimum uplift acceleration is a direct consequence of the multiple possible ways that a masonry arch with finite thickness may rupture at a given location. The paper further confirms that the results obtained with commercially available distinct element software are in very good agreement with the rigorous solution.     

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.     

A methodology to assess the rate and pressure sensitivity of polymers over a wide range of strain rates

This paper details a methodology to test the mechanical response of soft, pressure sensitive materials, over a wide range of strain rates. A hybrid experimental-numerical procedure is used to assess the constitutive parameters. The experimental phase involves axial compression of a cylindrical specimen which is confined by a tightly-fit sleeve that is allowed to yield plastically, thus applying a constant confining pressure. The usually neglected frictional effects between the specimen and the sleeve are fully accounted for and characterized in detail. With commercial polycarbonate as a typical example, it is shown that pressure sensitivity and rate sensitivity are not coupled, thus reducing the number of tests needed to characterize a material. The results of numerical simulations indicate that the pressure sensitivity index (angle β in the Drucker-Prager material model) has little influence on the hydrostatic and confining pressures, whereas the equivalent stress sustained by the specimen increases with β, which for commercial polycarbonate is found to be β=15°.     

Conjugate natural convection heat transfer between two fluids separated by a horizontal wall: steady-state analysis

The conjugate heat transfer across a thin horizontal wall separating two fluids at different temperatures is investigated both numerically and asymptotically. The solution for large Rayleigh numbers is shown to depend on two nondimensional parameters;α/ε ², withα being the ratio of the thermal resistance of the boundary layer in the hot medium to the thermal resistance of the wall andε the aspect ratio of the plate, andβ, the ratio of the thermal resistances of the boundary layers in the two media. The overall Nusselt number is an increasing function ofα/ε ² taking a finite maximum value forα/ε ² → ∞ and tending to zero forα/ε ² → 0.     

Dependence of flow classification on the Reynolds number for a two-cylinder wake

This work aims to investigate the dependence of flow classification on the Reynolds number (Re) for the wake of two staggered cylinders. The Re examined ranges from 1.5×10³ to 2.0×10⁴. The pitch ratio, P^⁎=P/d examined is 1.2–6.0 (d is the cylinder diameter), and angle (α) is 0–90°, where P is the center-to-center spacing between two cylinders and α is the angle between the incident flow and the line through the cylinder centers. Two single hotwires were used to measure simultaneously the fluctuating streamwise velocities (u) in the vortex streets generated by the two cylinders. The power spectral density functions and the Strouhal numbers were then obtained from the u signals, based on which the flow structure pattern or mode could be determined. Over two hundred configurations of two staggered cylinders have been examined for each Re. It is found that Re has an appreciable effect on the dependence of the flow mode on P^⁎ and α. The observation is connected to the Re effect on the generic features of a two-cylinder wake such as flow separation, boundary layer thickness, gap flow deflection and vortex formation length.     

Proposal of FEM implemented with particle discretization for analysis of failure phenomena

A new method for numerical simulation of failure behavior, namely, FEM-β, is proposed. For a continuum model of a deformable body, FEM-β solves a boundary value problem by applying particle discretization to a displacement field; the domain is decomposed into a set of Voronoi blocks and the non-overlapping characteristic functions for the Voronoi blocks are used to discretize the displacement function. By computing average strain and average strain energy, FEM-β obtains a numerical solution of the variational problem that is transformed from the boundary value problem. In a rigorous form, FEM-β is formulated for a variational problem of displacement and stress with different particle discretization, i.e., the non-overlapping characteristic function of the Voronoi blocks and the conjugate Delaunay tessellations, respectively, are used to discretize the displacement and stress functions. While a displacement field is discretized with non-smooth functions, it is shown that a solution of FEM-β has the same accuracy as that of ordinary FEM with triangular elements. The key point of FEM-β is the ease of expressing failure as separation of two adjacent Voronoi blocks owing to the particle discretization that uses non-overlapping characteristic functions. This paper explains these features of FEM-β with results of numerical simulation of several example problems.     

Quadrature formulas of maximum algebraic accuracy for cauchy type integrals with complex exponents of the Jacobi weight function

The present paper presents a Gauss type quadrature formula for a Cauchy type integral whose density is the product of a Hölder function by the weight function (1 ? x) ^α (1 + x) ^β (Re α, Reβ > ?1) of orthogonal Jacobi polynomials. It is shown that at the roots of the function of the second kind corresponding to the Jacobi polynomial P _n ^(α,β) (x), the quadrature formula with n nodes gives the exact value of a Cauchy type integral for an arbitrary polynomial of order k ≤ 2n. This formula was tested when solving several contact and mixed problems of the theory of elasticity.     

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when β_c<β<0; there is one solution for (β=β_c)∪(β≥0); there is no solution for β<β_c, where β_c is a critical unsteadiness parameter dependent on mass transfer parameter.     

Initiation of fracture at the interface corner of bi-material joints

Within the context of linear elasticity, a stress singularity of the form Hr^λ−1 may exist at the interface corner of a bi-material joint, where r is the radial distance from the corner, H is the stress intensity factor and λ−1 is the order of the singularity. Recent experimental results in the literature support the use of a critical value of the intensity factor H=H_c as a fracture initiation criterion at the interface corner. In this paper, we examine the validity and limitations of this criterion for predicting the onset of fracture in a butt joint consisting of a thin layer of an elastic-plastic adhesive layer sandwiched between two elastic adherends. The evolution of plastic deformation at the corner is determined theoretically and by the finite element method, and the solution is compared with the extent of the elastic singular field. It is shown that H_c is a valid fracture parameter if h>B(H_c/σ_Y)^1/(1−λ) where the non-dimensional constant B=100 for β=0 and B=13 for β=α/4. Here, h is the thickness of the adhesive layer, σ_Y is the uniaxial yield stress of the bulk adhesive and (α,β) are Dundurs’ parameters (Dundurs, J., J. Appl. Mech. 36 (1969) 650). Experimental results for aluminium/epoxy/aluminium and brass/solder/brass sandwiched joints are used to assess the role of plastic deformation on the validity of the failure criterion.     

Time-dependent stagnation-point flow over rotating disk impinging oncoming flow

The unsteady stagnation point flow of an incompressible viscous fluid over a rotating disk is investigated numerically in the present study.The disk impinges the oncoming flow with a time-dependent axial velocity.The three-dimensional axisymmetric boundary-layer flow is described by the Navier-Stokes equations.The governing equations are solved numerically,and two distinct similarity solution branches are obtained.Both solution branches exhibit different flow patterns.The upper branch solution exists for all values of the impinging parameter β and the rotating parameter.However,the lower branch solution breaks down at some moderate values of β.The involvement of the rotation at disk allows the similarity solution to be transpired for all the decreasing values of β.The results of the velocity profile,the skin friction,and the stream lines are demonstrated through graphs and tables for both solution branches.The results show that the impinging velocity depreciates the forward flow and accelerates the flow in the tangential direction.     

Natural convection in an inclined quadrantal cavity heated and cooled on adjacent walls

Natural convective flow and heat transfer in an inclined quadrantal cavity is studied experimentally and numerically. The particle tracing method is used to visualize the fluid motion in the enclosure. Numerical solutions are obtained via a commercial CFD package, Fluent. The working fluid is distilled water. The effects of the inclination angle, ? and the Rayleigh number, Ra on fluid flow and heat transfer are investigated for the range of angle of inclination between 0° ? ? ? 360°, and Ra from 10⁵ to 10⁷. It is disclosed that heat transfer changes dramatically according to the inclination angle which affects convection currents inside, i.e. flow physics inside. A fairly good agreement is observed between the experimental and numerical results.     

The water rise in front of a model planing hull

It is known that the water “splashes-up” or rises above the undisturbed surface immediately in front of a planing surface. This rise is greatest in front of a flat planing plate and a number of attempts have been made to reduce the experimental measurements of this phenomenon to some kind of order. Since it was first independently proposed by both Schnitzer and Smiley in 1952, all attempts to correlate the flat plate splash-up have started with the assumption that splash is only a function of the immersed length of the plate and is independent of trim angle at least below about 20°. In part, this was because the three early studies which compared this hypothesis with experimental data omitted those portions of the data which did not support the hypothesis. The present paper concludes that this forty year old hypothesis is fallacious and that the water rise in front of any prismatic planing surface is best approximated by $$\frac{d}{{\sqrt {bl} }} = k\sin ^2 {\mathbf{\tau }}$$ whered is the vertical water rise at the water/keel intersection;b is the beam;l is the submerged length of the keel;τ is the trim angle;k: is a constant determined from experiment, approximated by,k = 2e ^?2.5β , whereβ is the deadrise angle in radians. It might be thought that this is a slight contribution, of little practical import, but for one thing. Starting in the 1950's most towing tank experimenters in the United States abandoned the difficult measurement of model draft and obtained only the “actually wetted length” from underwater photographs. But theoretical planing force calculations require a knowledge of the relationship between a hull and the undisturbed water plane. Thus if modern experimental data is to be compared with theory, it is necessary to estimate what the undefined splash-up or water rise was during each experiment, in order to estimate the model's true position in space. The paper concludes by criticizing the format of some modern reports of experiments with model planing hulls and suggests that, in addition to the usual graphical presentations, measured data should always be reported numerically. Also, that when relevant data is omitted from a plot, the facts of such omission should be clearly stated.     

Analytical solutions for tapering quadratic and cubic rat-holes in highly frictional granular solids

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax², and three-dimensional cubic rat-holes of profile z=ar³, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.     

Time-averaged topological flow patterns and their influence on vortex shedding of a square cylinder in crossflow at incidence

Flow characteristics around the square cylinder and their influence on the wake properties are studied. Time-averaged flow patterns on the surfaces of square cylinder in a cross-stream at incidence are experimentally probed by surface-oil flow technique and analyzed by flow topology for Reynolds numbers between 3.9×10⁴ and 9.4×10⁴ as the incidence angle changes from 0° to 45°. Vortex shedding characteristics are measured by a single-wire hot-wire anemometer for Reynolds numbers between 5×10³ and 1.2×10⁵. The effects of topological flow patterns on the wake properties then are revealed and discussed. Flows around the square cylinder are identified as three categories: the subcritical, supercritical, and wedge flows according to the prominently different features of the topological flow patterns. The Strouhal number of vortex shedding, turbulence in the wake, and wake width present drastically different behaviors in different characteristic flow regimes. A critical incidence angle of 15° separates the subcritical and supercritical regimes. At the critical incidence angle the wake width and shear-layer turbulence present minimum values. The minimum wake width appearing at the critical incidence angle, which leads to the maximum Strouhal number, is due to the reattachment of one of the separated boundary layer to the lateral face of the square cylinder. If the Strouhal numbers are calculated based on the wake width instead of the cross-stream projection width of cylinder, the data in the subcritical and supercritical regimes are well correlated into two groups, which would approach constants at high Reynolds numbers.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n+α+β-dimensional averaged Fokker-Planck-Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, α combinations of angle variables and β combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has α internal resonant relations and the system and harmonic excitations have β external resonant relations. The averaged FPK equation is solved by using the combination of the finite difference method and the successive over relaxation method. Two coupled Duffing-van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined.     

The effects of the rock bridge ligament angle and the confinement on crack coalescence in rock bridges: An experimental study and discrete element method

The present article investigates the influences of the rock bridge ligament angle, β, and the confinement on crack coalescence patterns by conducting laboratory and numerical tests on rock-like specimens. Laboratory tests show that no coalescence in the rock bridge occurred for low β. With an increase of β, tensile-shear coalescence and tensile coalescences subsequently occurred. In addition, the increase in the confinement first promoted shear coalescence and then restrained crack coalescence for low β, whereas the tensile coalescence was restrained by the increase in confinement for high β. The numerical results corroborate the laboratory tests in the coalescence patterns. In addition, the numerical study shows that tensile and shear cracks subsequently initiated near crack tips because of the concentrated tensile and shear stresses, respectively. Regarding the influence of β on crack coalescence, tensile or shear stress failed to concentrate in rock bridges for low β. Therefore, the cracks failed to coalesce, whereas with the increase in β, tensile and shear stress concentrations occurred in the bridge and led to either tensile shear or tensile coalescence. Regarding the influence of confinement on crack coalescence, the increase in confinement restrained the tensile stress concentrations and further hindered tensile crack coalescence in rock bridges for high values of β.     

Analysis of upstream,double-row,cylindrical holes on primary and secondary effects of endwall flow and film cooling

Flow features and film cooling performance of five configurations of double-row, cylindrical holes, upstream of an E3 vane, in a linear cascade are numerically investigated. This simulation is completed using a verified turbulence model at four blowing ratios (M = 0.5, 1.0, 1.5, 2.0). The first three configurations have two rows of cylindrical holes, each row with the same compound angle (β=-45°, 0° or 45°), while the other two have two rows with opposite compound angles (β=-45°, 45° and β=45°, -45°), which are also referred to as double-jet film cooling (DJFC) holes. The primary effects on the downstream endwall and the secondary effects on the nearby airfoil of the cooled passage are analyzed and discussed in detail. Results show that at low blowing ratios the movement of the coolant is denominated by the interaction between the jets and vortices resulting in similar film coverage on both the endwall and airfoil. The effect of vortices is reduced at high blowing ratios. It is also shown that the movement of the coolant is determined by the initial velocity direction, as well as the film cooling configuration.