Showing papers in "Acta Mechanica in 2008"

Thirty years of numerical flow simulation in hydraulic turbomachines

Abstract: The application of computational fluid dynamics (CFD) in the design of water turbines and pumps started about 30 years ago. This paper reviews the main steps and breakthroughs in the methods that were made during this period, through the eyes of one particular water turbine company which spear-headed some of the first developments for practical applications. Practical examples are used to illustrate the developments of the tools from 1978 to 2008 and to give an overview of the complete revolution in hydraulic turbine design that has occurred over this time. Several periods with distinct levels of complexity, and hence accuracy of the physical models and of the simulation methods can be distinguished. The first steps coincided with the introduction of the Finite Element Method into CFD, and were characterized by simplified Quasi-3D Euler solutions and Fully 3D potential flow solutions. Over the years the complexity continuously increased in stages: via 3D Euler solutions, to steady RANS simulations of single blade passages using finite volume methods, extending to steady simulations of whole machines, until today unsteady RANS equations are solved with advanced turbulence models. The most active areas of research and development are now concerned with including the effects of 2-phase flows (free surface flow in Pelton turbines and cavitation) and fluid–structure interaction.

On new erosion models of Savage–Hutter type for avalanches

Abstract: In this work, we study the modeling of one-dimensional avalanche flows made of a moving layer over a static base, where the interface between the two can be time dependent. We propose a general model, obtained by looking for an approximate solution with constant velocity profile to the incompressible Euler equations. This model has an energy dissipation equation that is consistent with the depth integrated energy equation of the Euler system. It has physically relevant steady state solutions, and, for constant slope, it gives a particular exact solution to the incompressible hydrostatic Euler equations. Then, we propose a simplified model, for which the energy conservation holds only up to third-order terms. Its associated eigenvalues depend on the mass exchange velocity between the static and moving layers. We show that a simplification used in some previously proposed models gives a non-consistent energy equation. Our models do not use, nor provide, any equation for the moving interface, thus other arguments have to be used in order to close the system. With special assumptions, and in particular small velocity, we can nevertheless obtain an equation for the evolution of the interface. Furthermore, the unknown parameters of the model proposed by Bouchaud et al. (J Phys Paris I 4,1383–1410, 1994) can be derived. For the quasi-stationary case we compare and discuss the equation for the moving interface with Khakhar’s model (J Fluid Mech 441,225–264, 2001).

Non-probabilistic set-theoretic model for structural safety measure

Abstract: In this paper, a new non-probabilistic set-theoretic safety measure for structures is proposed. Based on the non-probabilistic set-theoretic stress–strength interference model, the ratio of the volume of the safe region to the total volume of the region associated with the variation of the basic interval variables is suggested as the measure of structural non-probabilistic safety. The compatibility between the presented non-probabilistic set-theoretic safety measure and the probabilistic reliability is demonstrated. Numerical examples are used to shed a light on the validity of the presented measure.

Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects

Abstract: We consider the macroscopic behavior of two-phase fibrous piezoelectric composites. The fibers are of circular cross-section with the same radius. Along the interfaces between the fibers and the matrix we consider the effects of surface stress and surface electric displacement. The constituents are transversely isotropic and exhibit pyroelectricity. We find that the overall thermoelectroelastic moduli of these solids must comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides five constraints between the six axisymmetric overall electroelastic moduli. The second set relates the effective coefficients of thermal stress and pyroelectric coefficients to the effective electroelastic moduli, in analogy with Levin’s formula. In contrast to their conventional counterparts, i.e., without surface effects, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.

A new model of granular flows over general topography with erosion and deposition

Abstract: A fundamental issue for describing gravity-driven flows over general topography is the search for an “optimal” coordinate. Bouchut and Westdickenberg [1] proposed an arbitrary coordinate system (BW) for general topography. The unified coordinate (UC) system (e.g., [2], [3]), which was developed for computational fluid dynamics, combines the advantages of both Eulerian and Lagrangian systems, so that the coordinates can instantaneously move with some singular surface within the flows. By utilizing the benefit of the BW coordinates and UC system, a new model of gravity-driven flows over general topography is derived, in which the erosion and deposition processes at the bed are considered. The depth-integrated mass and momentum equations are presented in the time-dependent and terrain-following coordinate system, which coincides with the interface distinguishing between the static and flowing layers. A shock-capturing numerical scheme is implemented to solve the derived equation system. Simulation results present the new features of this model and reveal a new physical insight of the erosion/deposition processes.

The role of air entrainment on the outcome of drop impact on a solid surface

Abstract: The characteristic conditions causing spreading or splashing after drop impact on solid surfaces are considered together with the underlying mechanisms To this end, the results of the various studies published over the past few years that have addressed the issue of splashing after droplet impact, specifically in terms of the definition of a splashing threshold, are critically compared and synthesized The discussion aims at clarifying some of the conflicting findings Information drawn from these considerations is used to distinguish between various splashing thresholds and it is shown that there exists a distinct difference between splashing on smooth and on rough surfaces, both in terms of the splashing thresholds and in terms of the mechanisms Finally, a physical mechanism akin to air entrainment in dynamic wetting is proposed that may be of primary importance for the inception of splashing as well as fingering on smooth surfaces

Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus

Abstract: This paper is concerned with the problem of a functionally graded coated half-space indented by an axisymmetric smooth rigid punch. The shear modulus of the graded coating is assumed to be an exponential function and the Poisson’s ratio is a constant. With the use of Hankel integral transform technique, the axisymmetric frictionless contact problem is reduced to a Cauchy singular integral equation. The contact pressure, contact radius and penetration depth are calculated for various indenters by solving the equations numerically. The results show that these quantities are greatly affected by the gradient of the coating.

Reformulation of mindlin–reissner governing equations of functionally graded circular plates

Abstract: The governing equations of the first-order shear deformation plate theory for FG circular plates are reformulated into those describing the interior and edge-zone problems. Analytical solutions are obtained for axisymmetric and asymmetric behavior of functionally graded circular plates with various clamped and simply-supported boundary conditions under mechanical and thermal loadings. The material properties are graded through the plate thickness according to a power–law distribution of the volume fraction of the constituents. The results, which are in closed form and suitable for design purposes, are verified with known results in the literature. It is shown that there are two boundary-layer equations. The effects of material property, plate thickness, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. Under a mechanical load, the responses of FG solid circular plates with various clamped supports are seen to be identical. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effects in clamped FG plates being stronger than those in simply-supported plates. Also an exact solution is developed for the one-dimensional heat conduction equation with variable heat conductivity coefficient.

Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect

Abstract: A class of nonsingular yield conditions depending on three parameters is analyzed for isotropic materials exhibiting strength differential effect and pressure insensitivity. The yield condition can then be expressed in terms of the second and third stress deviator invariants. The convexity requirement is considered and the constraints imposed on the material parameters are discussed in detail. The dual dissipation function is derived in the analytical form. The condition can be applied in the analysis of high strength alloys (such as Inconnel 718) or of shape memory alloys (such as NiTi, NiAl, CuZnGa, or CuAlNi) in order to specify the onset of yield, or of martensitic or austenitic transformation. The conditions can easily be generalized to account for mixed hardening and back stress anisotropy. Some experimental data are provided to verify the proposed conditions.

Transient Couette flow in a rotating non-Darcian porous medium parallel plate configuration : network simulation method solutions

Abstract: The transient, viscous, incompressible, hydrodynamic Couette flow in a rotating porous medium channel is studied in this paper. The channel comprises a pair of infinitely long parallel plates which rotate with uniform angular velocity about an axis normal to the plates. The porous medium is simulated using a Darcy–Forchheimer drag force model which includes both bulk matrix porous drag (dominant at low Reynolds numbers) and second order inertial impedance (dominant at higher Reynolds numbers). The two-dimensional Navier–Stokes equations are reduced to a (z*, t*) coordinate system incorporating Coriolis terms, and appropriate initial and boundary conditions are prescribed. Separate porous drag body force terms are incorporated in both the primary and secondary flow momentum equations. Using a set of transformations, the model is rendered dimensionless and shown to be dictated by the Ekman number, Forchheimer number, Darcy number and Reynolds number in a (z, t) coordinate system. Numerical solutions are obtained for the transformed model using the Network Simulation Method. The influence of the hydrodynamic parameters are computed graphically and also the interaction of parameters on the velocity fields is discussed at length. Excellent agreement is found with earlier non-porous flow studies. The analysis has important applications in geophysics and also chemical engineering systems.

Enhanced damping of lightweight structures by semi-active joints

Abstract: Lightweight structures typically have low inherent structural damping. Effective vibration suppression is required, for example, in certain applications involving precision positioning. The present approach is based on friction damping in semi-active joints which allow relative sliding between the connected parts. The energy dissipation due to interfacial slip in the friction joints can be controlled by varying the normal pressure in the contact area using a piezo-stack actuator. This paper focuses on the optimal placement of semi-active joints for vibration suppression. The proposed method uses optimality criteria for actuator and sensor locations based on eigenvalues of the controllability and observability gramians. Optimal sensor/actuator placement is stated as a nonlinear multicriteria optimization problem with discrete variables and is solved by a stochastic search algorithm. At optimal locations, conventional rigid connections of a large truss structure are replaced by semi-active friction joints. Two different concepts for the control of the normal forces in the friction interfaces are implemented. In the first approach, each semi-active joint has its own local feedback controller, whereas the second concept uses a global, clipped-optimal controller. Simulation results for a 10-bay truss structure show the potential of the proposed semi-active concept.

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom

Abstract: We show that symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of finite isotropic hyperelasticity with exact rotational degrees of freedom. This is contrary to claims in the literature which are valid, however, in the linear isotropic case.

The physics of unsteady jet impingement and its heat transfer performance

Abstract: The influence of unsteadiness with respect to the heat transfer performance of impinging jets is studied systematically by imposing various shapes and frequencies of unsteadiness. By means of an enhancement coefficient (Nusselt number of the unsteady flow over that for the steady flow) the heat transfer performance is characterized and most efficient types of unsteadiness are identified. From PIV data details of the flow field emerge. These data help to better understand the physics of heat transfer enhancement. Finally it is shown that the influence of unsteadiness can be very different when small obstacles are added to the smooth heat transfer surface, which corresponds to a more realistic situation.

Attitude dynamics and control of satellites orbiting rotating asteroids

Abstract: The paper focuses on the attitude dynamics and control of satellites orbiting a rotating asteroid. The general formulation of the satellite equations of motion in an equatorial eccentric orbit is obtained through the Lagrangian method. The linearized system model is derived and the stability analysis is presented. The control laws for three-axis attitude control of satellites are developed and a closed-form solution of the system is derived. For an illustration of the linear analysis followed by the numerical simulation of the governing nonlinear equations of motion of the satellite, several asteroids (Eros, Castalia, Vesta, Ida and Gaspra) are considered. Attitude resonances for satellites in retrograde orbits are found. The results of the linear system model compare well with the corresponding system nonlinear equations of motion. The proposed controllers are successful in stabilizing the attitude of the satellites even in the presence of high attitude disturbances and orbital eccentricities.

Appraisal of energy recovering sub-grid scale models for large-eddy simulation of turbulent dispersed flows

Abstract: Current capabilities of Large-Eddy Simulation (LES) in Eulerian-Lagrangian studies of dispersed flows are limited by the modeling of the Sub-Grid Scale (SGS) turbulence effects on particle dynamics. These effects should be taken into account in order to reproduce accurately the physics of particle dispersion since the LES cut-off filter removes both energy and flow structures from the turbulent flow field. In this paper, we examine the possibility of including explicitly SGS effects by incorporating ad hoc closure models in the Lagrangian equations of particle motion. Specifically, we consider candidate models based on fractal interpolation and approximate deconvolution techniques. Results show that, even when closure models are able to recover the fraction of SGS turbulent kinetic energy for the fluid velocity field (not resolved in LES), prediction of local segregation and, in turn, of near-wall accumulation may still be inaccurate. This failure indicates that reconstructing the correct amount of fluid and particle velocity fluctuations is not enough to reproduce the effect of SGS turbulence on particle near-wall accumulation.

Erratum to: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate

Abstract: Exact solutions for the unsteady flow of an Oldroyd-B fluid produced by a suddenly moved plane wall between two side walls perpendicular to the plane are established by means of the Fourier sine transforms The similar solutions for Maxwell, Newtonian and second grade fluids, performing the same motion, appear as limiting cases of the solutions obtained here In the absence of the side walls, the solutions corresponding to the motion over an infinite suddenly moved plate are also obtained as the limiting cases Finally, for comparison, the velocity field in the middle of the channel and the shear stress at the bottom wall are plotted for different values of the material constants

Observations on the definition of yield stress

Abstract: The difficulties of establishing an unambiguously physical definition for the yield stress property are discussed. Although too complex for routine application, a specific and completely physical definition of yield stress is developed and illustrated here, showing that the definition need not be of an arbitrary specification.

Steady-state creep of a pressurized thick cylinder in both the linear and the power law ranges

Abstract: The classical solution of the steady-state creep problem for a pressurized thick-walled cylinder is based on the power law constitutive equation. Several heat resistant steels show, however, the linear dependence of the creep rate on the applied stress within a certain stress range. In this paper we apply an extended constitutive equation which includes both the linear and the power law stress dependencies. The material constants are identified for the 9Cr1MoVNb steel at 600 °C. We recall the boundary value problem of steady-state creep for the thick cylinder under the plane strain condition. We present an approximate solution illustrating the stress redistributions as a result of the creep process. The analysis shows that for the certain range of the internal pressure both the linear and the power law creep must be taken into account. In this case the results according to the extended constitutive model essentially differ from the classical ones. The obtained solution is also applied to verify the developed user-defined creep material subroutine inside a commercial finite element code.

Direct Numerical Simulation of Turbulent Heat Transfer Modulation in Micro-Dispersed Channel Flow

Abstract: The objective of this paper is to study the influence of dispersed micrometer size particles on turbulent heat transfer mechanisms in wall-bounded flows. The strategic target of the current research is to set up a methodology to size and design new-concept heat transfer fluids with properties given by those of the base fluid modulated by the presence of dynamically-interacting, suitably-chosen, discrete micro- and nano-particles. We ran direct numerical simulations for hydrodynamically fully developed, thermally developing turbulent channel flow at shear Reynolds number Re τ = 150 and Prandtl number Pr = 3, and we tracked two large swarms of particles, characterized by different inertia and thermal inertia. Preliminary results on velocity and temperature statistics for both phases show that, with respect to single-phase flow, heat transfer fluxes at the walls increase by roughly 2% when the flow is laden with the smaller particles, which exhibit a rather persistent stability against non-homogeneous distribution and near-wall concentration. An opposite trend (slight heat transfer flux decrease) is observed when the larger particles are dispersed into the flow. These results are consistent with previous experimental findings and are discussed in the frame of the current research activities in the field. Future developments are also outlined.

Cable vibration mitigation by added SMA wires

Abstract: Limiting the oscillations of suspension or cable-stayed bridges requires a suitable strategy for the cable vibration mitigation. This paper approaches the problem by a campaign of laboratory tests conducted for a suspended steel cable specimen of diameter 2 mm. First the dynamic behaviour of the cable in free vibrations is studied. The main mechanical parameters, such as natural frequencies and damping coefficients, are evaluated by an elaboration of the time histories recorded by two tri-axial accelerometers. A wire of diameter 1 mm, made by shape memory alloy (SMA), is then added to the cable. Several configurations for the system “steel cable–SMA wire” are conceived. The goal is to find how the dynamic behaviour of the steel cable changes, and to assess which configuration could give the best results in terms of cable vibration mitigation and damping increase.

Modeling cementitious materials as multiphase porous media: theoretical framework and applications

Abstract: In this paper a general model for the analysis of concrete as multiphase porous material, based on the so-called Hybrid Mixture Theory, is presented. The development of the model equations, taking into account both bulk phases and interfaces of the multiphase system is described, starting from the microscopic scale. An exploration of the second law of thermodynamics is also presented: it allows defining several quantities used in the model, like capillary pressure, disjoining pressure or effective stress, and to obtain some thermodynamic restrictions imposed on the evolution equations describing the material deterioration. Then, two specific forms of the general model adapted to the case of concrete at early ages and beyond and to the case of concrete structures under fire are shown. Some numerical simulations aimed to prove the validity of the approach adopted also are presented and discussed.

Thermal shock analysis for a functionally graded plate with a surface crack

Abstract: The fracture behavior of a functionally graded material (FGM) plate subjected to a thermal shock is studied. A surface crack is considered. The thermomechanical properties of the FGM plate are assumed to vary along the thickness direction. By using a perturbation method, the transient temperature field is solved. Then the transient thermal stresses and the corresponding thermal stress intensity factor (TSIF) are obtained. The transient thermal stresses and TSIF in an FGM ceramic/metal (ZrO2/Ti-6Al-4V) plate are shown in figures.

On dispersion relations of Rayleigh waves in a functionally graded piezoelectric material (FGPM) half-space

Abstract: For propagation of Rayleigh surface waves in a transversely isotropic graded piezoelectric half-space with material properties varying continuously along depth direction, the Wentzel–Kramers–Brillouin (WKB) technique is employed for the asymptotic analytical derivations. The phase velocity equations for both the electrically open and shorted cases at the free surface are obtained. Influences of piezoelectric material parameters graded variations on Rayleigh wave dispersion relations, particles’ displacements magnitude and corresponding decay properties are discussed. Results obtained indicate that coupled Rayleigh waves can propagate at the surface of the graded piezoelectric half-space, and their dispersion relations and the particles displacements ellipticity at the free surface are dependent upon the graded variation tendency of the material parameters. By the Rayleigh surface waves phase velocities relative changing values combined with the relationship between the wave number and the material graded coefficient, a theoretical foundation can be provided for the graded material characterization by experimental measurement.

A finite element analysis of frequency–temperature relations of AT-cut quartz crystal resonators with higher-order Mindlin plate theory

Abstract: The frequency–temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes are strongly coupled, has been an important requirement in most applications. The analytical work on frequency–temperature relations has been done over the last decades in many aspects, ranging from the fundamental theory for the thermal effect on vibrations of elastic solids to simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to complications of the resonator structures such as the presence of a mounting structure and electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects. In this paper, we incorporate the frequency–temperature theory for crystal plates based on the incremental thermal field formulation by Lee and Yong into our finite element analysis implementation, which is then used to analyze the free vibrations of crystal plates with the higher-order Mindlin plate theory. The effect of electrodes on the frequency–temperature relation is also considered. The computational results are compared with experimental ones from actual products. The satisfactory agreement demonstrates the precise prediction of the frequency–temperature behavior and practical applications of the finite element analysis in product modeling and development.

MHD boundary-layer flow of a micropolar fluid past a wedge with variable wall temperature

Abstract: The steady laminar MHD boundary-layer flow past a wedge immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results show that the micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.

About a model of structural-phase transformations under hydrogen influence

Abstract: Hydrogen embrittlement of materials is one of the important factors for an estimation of serviceability of designs. In spite of the fact that many works are devoted to research of hydrogen influence on properties of metals and alloys, there are enough “white” spots in this problem, in particular, the influence of kinetic processes inside a material, giving rise to hydrogen redistribution, on the basic strength characteristics at static and dynamic loading. The absence of an authentic mechanical model, whose description is based on the basic fundamental principles of rational mechanics, makes the problem rather actual, since, inherently, the presence of such a model would allow to describe the influence of internal kinetics on the macroparameters of a material, which is today extremely important in view of the huge amount of hypotheses which are ready to confirm only special experiments and do not allow to use them in other cases. The present work offers a model based on experimental data on the decrease in break energy of internuclear bonds at landing particles of hydrogen, i.e., the decrease in a level of the free energy. The offered model allows to explain the localization of hydrogen in the field of the increased concentration of stresses and change of material properties.

An asymptotic description of the elastic instability of twisted thin elastic plates

Abstract: The work presented here reconsiders the classical stability problem for the deformation experienced by a stretched elastic strip when its ends are subjected to small twisting moments. Singular perturbation methods enable us to describe analytically the wrinkling instability that occurs when the strip is very thin. In this case the localised structure of the instability pattern is controlled by the solution of a second-order boundary value problem with variable coefficients. The theoretical results obtained are confirmed by direct numerical simulations of the full problem.

Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Abstract: The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.

A cracked functionally graded piezoelectric material strip under transient thermal loading

Abstract: Considered in this paper is a functionally graded piezoelectric material strip (FGPM strip) containing an embedded crack or an edge crack perpendicular to its boundaries. The problem is solved for an FGPM strip that is suddenly heated or cooled from the bottom surface under static mechanical loading. The top surface is maintained at the initial temperature. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the bottom surface. First, the transient temperature and the thermal stress distributions in an uncracked strip are calculated by using the Laplace transform. Then, these mechanical and thermal stresses are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. By using the Fourier transform, the electromechanical problem is reduced to a singular integral equation which is solved numerically. The numerical results for the stress and thermal stress intensity factors are computed as a function of the normalized time, the nonhomogeneous and geometric parameters. The temperature and the thermal stress distributions for the uncracked problem and the results for the crack contact problem are also included.

A three-phase cylindrical shear-lag model for carbon nanotube composites

Abstract: In this article the viscoelastic behavior of carbon nanotubes (CNTs) reinforced composites is investigated theoretically by using the three-phase concentric cylindrical shell model along with shear-lag arguments. The parameters which influence the fiber stress, the matrix stress and the interfacial stress have been revealed. The aspect ratio of CNTs βt, the cross-sectional area ratio of CNTs βA, the matrix-to-fiber modulus ratio λm and the interphase-to-fiber modulus ratio λn are common influencing parameters of both the stresses in nanocomposites and the composite modulus. In addition, the effective composite modulus has three other influencing parameters of its own, i.e., the fiber volume fraction vf, the interphase volume fraction vn and the RVE-to-fiber length ratio η, whereas the stresses have their own influencing parameters of the RVE-to-fiber diameter ratio βR and the interphase-to-fiber diameter ratio βb. The modulus of CNTs composites depends strongly upon the modulus and thickness of the interphase. Carbon nanotube fibers improve the viscoelastic stiffness in the whole time period. However, the magnitude of modulus improvement does not vary monotonically with time.