Abstract: A Mixed Finite Element (MFE) method for 3D non-steady flow of a viscoelastic compressible fluid is presented. It was used to compute polymer injection flows in a complex mold cavity, which involves moving free surfaces. The flow equations were derived from the Navier-Stokes incompressible equations, and we extended a mixed finite element method for incompressible viscous flow to account for compressibility (using the Tait model) and viscoelasticity (using a Pom-Pom like model). The flow solver uses tetrahedral elements and a mixed velocity/pressure/extra-stress/density formulation, where elastic terms are solved by decoupling our system and density variation is implicitly considered. A new DEVSS-like method is also introduced naturally from the MINI-element formulation. This method has the great advantage of a low memory requirement. At each time slab, once the velocity has been calculated, all evolution equations (free surface and material evolution) are solved by a space-time finite element method. This method is a generalization of the discontinuous Galerkin method, that shows a strong robustness with respect to both re-entrant corners and flow front singularities. Validation tests of the viscoelastic and free surface models implementation are shown, using literature benchmark examples. Results obtained in industrial 3D geometries underline the robustness and the efficiency of the proposed method
Abstract: An efficient method to compute permeability of fibrous media is presented. An immersed domain approach is used to represent the porous material at its microscopic scale and the flow motion is computed with a stabilized mixed finite element method. Therefore the Stokes equation is solved on the whole domain (including solid part) using a penalty method. The accuracy is controlled by refining the mesh around the solid-fluid interface defined by a level set function. Using homogenisation techniques, the permeability of a representative elementary volume (REV) is computed. The computed permeabilities of regular fibre packings are compared to classical analytical relations found in the bibliography
Abstract: In this paper, a numerical approach for permeability determination at the mesoscopic and microscopic scales is proposed. In an eulerian framework, the computational domain (corresponding to the elementary cell) is composed of one single mesh, where the interface between the fibres (microscopic scale) or the yarns composing the fabric (mesoscopic scale) and fluid is captured through a level set approach. At the microscopic scale, Stokes equations are considered. At the mesoscopic scale, resolution of a coupled Stokes (in the fluid)Darcy (in the yarn) flow is necessary and is performed using also a mixed finite element technique, providing a single system of equations. Stabilization of the Brinkman flow is attained using the P1+/P1 element. Results on permeability computation at the microscopic scale, as well as sensitivity analysis, illustrate the methodology followed.
Abstract: This paper focuses on improving the description of the contact between solid particles in a fluid flow. The numerical approach used is related to the fictitious domain method for the fluid'solid problem. It is associated to a gluey particle model in order to improve the behaviour of the particles during their contacts as a Lagrangian method is applied for their displacement. The numerical methodology is validated through 2D and 3D computations describing interactions of two particles in a shear flow. The results obtained show the ability of the scheme to recover the reversibility of the Stokes equations, even for 3D configurations. Finally, another example is studied with larger number of particles.
Abstract: Coextrusion technologies are commonly used to produce multilayered composite sheets or films for a large range of applications from food packaging to optics. The contrast of rheological properties between layers can lead to interfacial instabilities during flow. Important theoretical and experimental advances regarding the stability of compatible and incompatible polymers have, during the last decades, been made using a mechanical approach. However, few research efforts have been dedicated to the physicochemical affinity between the neighboring layers. The present study deals with the influence of this affinity on interfacial instabilities for functionalized incompatible polymers. Polyamide (PA6)/polyethylene grafted with glycidyl methacrylate (PE-GMA) was used as a reactive system and PE/PA6 as a non reactive one. Two grades of polyamide (PA6) were used in order to change the viscosity and elasticity ratios between PE (or PE-GMA) and PA6. It was experimentally confirmed, in this case, that weak disturbance can be predicted by considering an interphase of non-zero thickness (corresponding to an interdiffusion/reaction zone) instead of a purely geometrical interface between the two reactive layers. According to rheological investigations from previous work, an experimental strategy was here formulated to optimize the process by listing the parameters that controlled the stability of the reactive multilayer flows. Plastic films with two layers were coextruded in symmetrical and asymmetrical configurations in which PA6 was the middle layer. Indeed, for reactive multilayered systems, the interfacial flow instability could be reduced or eliminated, for instance, by (i) increasing the residence time or temperature in the coextrusion bloc (for T above the reaction temperature Tâ=â240°C), and (ii) reducing the total extrusion flow rate. The reaction rate/compatibilization played a major role that must be taken into account. Furthermore, the role of the viscosity ratio, elasticity ratio, and layer ratio of the stability of the interface were also investigated coupling to the physicochemical affinity. The results show that it is necessary to obtain links between the classic factors that are introduced in the evaluation of the theoretical, given by linear stability analysis/longwave asymptotic investigations, and its corresponding experimental stability charts. Hence, based on this analysis, guide-lines for a stable coextrusion of reactive functionalized polymers can be provided.
Abstract: The flow motion of solid particle suspensions is a fundamental issue in many problems of practical interest. The velocity field of a such system is computed by a finite element method with a multi-domain approach of two phases (namely a viscous fluid and rigid bodies), whereas the particle displacement is made by a particulate method. We focus our paper on a simple shear flow of Newtonian fluid.
Abstract: A fluid-structure interaction method, based on a eulerian monolithic approach is introduced in order to study distributive and dispersive aspects of mixing within a multiscale approach of the processes. A first example investigates macroscopic flow resolution for the whole process including moving tools, then the dispersion of a single agglomerate is studied within a microscopic approach, and finally one shows a full macroscopic simulation coupled through a kinetic theory for dispersive mixing.
Notes: 18th Congress of French Mechanics, Grenoble, FRANCE, AUG 27-31, 2007
Abstract: Experimental Investigation of the Development of Interfacial Instabilities in Two Layer Coextrusion Dies
The stability of two-layer flow of polyethylene and polystyrene is experimentally studied in different flow geometries and for various flow rate ratios. A first coextrusion device allows to stop the coextrusion flow in a very long slit channel, to cool down the polymer sample and to dismantle the die in order to extract extrudate which is then carefully analyzed. A second device allows to observe the whole slit flow through transparent lateral walls and to record the interfacial waves in both spontaneous and controlled unstable conditions. Both devices point clearly out that the interfacial defect begins to grow after a specific flow distance and is then quickly amplified. This demonstrates the convective character of interfacial wave. Controlled unstable processing conditions in transparent die allow to measure accurately growth rate of defect in the linear regime and show quick occurrence of non linear regime.
Abstract: The interface instability of the coextrusion flow of a polyethylene and a polystyrene is studied both experimentally and theoretically in a slit geometry. For prototype industrial conditions, we have found a stable/unstable transition which bounds the occurrence of stable/unstable sheets at die exit. By investigating a large range of processing conditions, we have shown that this transition is controlled by both temperature and flow rate ratios. Close to the transition, we used a transparent die to measure spatial amplification of different controlled perturbations at die inlet and pointed out the convective nature of the instability which exhibits a dominant mode (for which the instability is the most severe). We have then found that a convective stability analysis, using the White-Metzner constitutive equation, is able to account for the spatial amplification rate experimentally measured on controlled perturbation experiments. By considering that the instability is controlled by its dominant mode, we performed a convective stability analysis for all studied prototype industrial conditions and showed that such an analysis is able to forecast the occurrence of defects at die exit. (C) 2004 Elsevier B.V. All rights reserved.
Abstract: The interface instability of the coextrusion flow of a polyethylene and a polystyrene is experimentally studied with industrial and laboratory equipments for various flow rates and temperature. Stable and unstable coextrusion conditions are identified as a function of flow rate ratio, shear rate and temperature. It is found that temperature and flow rate ratio are the most relevant parameters. Experimental results are then compared with stability analysis assuming White-Metzner constitutive equations. Agreement is fair at temperatures of 180°C and 200°C. However the longwave stability analysis is not sufficient to predict all experimental data.
Abstract: This paper is devoted to the numerical study of the stability of the plane flow for two immiscible fluids obeying Maxwell constitutive equation in a channel of aspect ratio 8. For fixed viscosity and flow rate ratios, the shape and non-linear behavior of interfacial waves induced by small perturbations of the input flow rate are studied using finite element simulations for various elasticity stratifications. The stationary flow, close to a Poiseuille flow and presenting a flat interface is first computed. Then the interfacial wave generated by a pulse-like disturbance on entry flow rates is computed by using time dependent computations. It takes the shape of a wave pocket which is amplified or damped while moving towards the die exit. In unstable cases, the maximum deviation increases with elasticity (the Weissenberg number). Interestingly, the elastic stratification between the two layers can counteract unstable waves due to viscous stratifications.
Abstract: The linear stability of three-layer plane Poiseuille flow is studied in the longwave limit and for moderate wavelengths. The fluids are assumed to follow Oldroyd-B constitutive equations with constant viscosities and elasticities. We find that the jumps of the Poiseuille shear rate at both interfaces which give the convexity of the Poiseuille velocity profile, allow us to determine the longwave stability for Newtonian fluids. On the other hand, the stability of viscoelastic fluids is analyzed by using the additive character of the longwave eigenvalues with respect to viscous and elastic terms. The stability with respect to moderate wavelength disturbances has to deal with two different modes called âshortwaveâ (SW) and âlongwaveâ (LW), according to their values at zero wavenumber. The SW eigenvalues can become the most dangerous modes for large Weissenberg numbers and their influences can be studied by means of shortwave analysis. Moreover, we point out that the longwave stability analysis and convexity of the Poiseuille velocity profile allow us to determine the LW eigenvalues which are stable with respect to order one wavelength disturbances. (C) 1999 Elsevier Science B.V. All rights reserved.
Abstract: The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier-Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets-associated with the full Navier-Stokes equations with viscous free-surface boundary conditions-are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, omega) = 0, where k is a wavenumber, omega is a frequency and D(k, omega) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier-Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability-as predicted by some model equations-is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem. The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.
Abstract: The convective dynamo is the generation of a magnetic field by the convective motion of an electrically conducting fluid. We assume a spherical domain and spherically invariant basic equations and boundary conditions. The initial state of rest is then spherically symmetric. A first instability leads to purely convective flows, the pattern of which is selected according to the known classification of O(3)-symmetry-breaking bifurcation theory. A second instability can then lead to the dynamo effect. Computing this instability is now a purely numerical problem, because the convective how is known only by its numerical approximation. However, since the convective how can still possess a nontrivial symmetry group G(0), this is again a symmetry-breaking bifurcation problem. After having determined numerically the critical linear magnetic modes, we determine the action of G(0) in the space of these critical modes. Applying methods of equivariant bifurcation theory, we can classify the pattern selection rules in the dynamo bifurcation. We consider various aspect ratios of the spherical fluid domain, corresponding to different convective patterns, and we are able to describe the symmetry and generic properties of the bifurcated magnetic fields.
Abstract: We study the double diffusion problem of a fluid confined between two spheres which are maintened at different temperature and salinity. The bifurcation parameters are the thermal Rayleigh number and the saline Rayleigh number. The linear stability analysis of the basic state of rest shows the existence, in the parameter plane, of a point at which the critical eigenvalue is non semisimple, with multiplicity 2(2l(c) + 1), where l(c) is a positive integer which is determined by the physical conditions on the problem, in particular the aspect ratio of the domain. We study this codimension two bifurcation point by exploiting the symmetries and normal form theory. We show that in the simplest case, i.e. l(c) = 1, the problem reduces to the case of a codimension two bifurcation with O(2) symmetry.
Abstract: The linear stability of plane Poiseuille flows of two and three-symmetrical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equations and hence the stability is controlled by the viscous and elastic stratifications and the layer thicknesses. For the three symmetrical-layer Poiseuille flow, competition between varicose (symmetrical) and sinuous (antisymmetrical) mode is considered. In both cases (two and three symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous and elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille flow is also stable with respect to moderate wavelength disturbances and the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity profile means a positive jump of shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important influence of the thickness ratio on the critical wavenumber is found for three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio). (C) 1997 Elsevier Science B.V.
Abstract: The linear and weakly nonlinear stability of flow in the Taylor-Dean system is investigated. The base flow far from the boundaries, is a superposition of circular Couette and curved channel Poiseuille flows. The computations provide for a finite gap system, critical values of Taylor numbers, wave numbers and wave speeds for the primary transitions. Moreover, comparisons are made with results obtained in the small gap approximation. It is shown that the occurrence of oscillatory nonaxisymmetric modes depends on the ââanisotropyâ coefficient in the dispersion relation, and that the critical Taylor number changes slightly with the azimuthal wave number for large absolute values of rotation ratio. The weakly nonlinear analysis is made in the framework of the Ginzburg-Landau equations for anisotropic systems. The primary bifurcation towards stationary or traveling rolls is supercritical when Poiseuille component of the base flow is produced by a partial filling. An external pumping can induce a subcritical bifurcation for a finite range of rotation ratio. Special attention is also given to the influence of anisotropy properties on the phase dynamics of bifurcated solution (Eckhaus and Benjamin-Feir conditions).
Abstract: This paper deals with the importance of an axial flux on the primary bifurcation of the Couette-Taylor system. Firstly, we explain to introduce a zero axial mean flow condition into the functional framework allowing us to compute numerically the amplitude equation. The new results yield excellent quantitative agreement with experimental and numerical simulation results above criticality. Secondly, we look at the influence of an additional small axial mean flow. This effect is treated as a perturbation to the classical situation. At small Reynolds number, this imperfection induces a Poiseuille flow in the axial direction. On increasing the Reynolds number, the transition to the stationary Taylor vortex is replaced by the appearance of a travelling wave with a group velocity two orders of magnitude larger,than the Poiseuille velocity. In the case when the oscillatory non-axisymmetric modes are the most unstable, the standing waves are now quasiperiodic and the two travelling waves have slightly different frequencies and are non symmetric. Moreover, the primary stable solution which bifurcates from the Couette flow is always the travelling wave moving within the Poiseuille flow, and there are parameter ranges where the wave which travels in the opposite direction to the main flux may be stable together with the other wave. The quasiperiodic solution can occur via a secondary bifurcation as the Reynolds number increases.
Abstract: We consider the classical Couette-Taylor problem in the limiting case when the radii ratio is very close to 1 in the case when the cylinders are counter-rotating. We propose a new limit system governing the perturbation, pointing out that in this case, the commonly used equations are incorrect. The new dimensionless parameters are now the two Taylor numbers and the basic flow is the Planar Couette flow. A linear analysis indicates the possibility of oscillatory instabilities. Bifurcating patterns are studied by means of the Ginzburg-Landau equations, and the novelty comes from the anisotropic properties which add advection and cross-derivative terms. The present work gives the numerical coefficients of these equations, including the singular cases of subcritical bifurcations of Taylor vortices and of the competition between steady and oscillatory modes.
Abstract: Flows involving natural convection are generally studied employing the Boussinesq approximation of the Navier-Stokes equations. This paper deals with large departures from this limit by considering the two-dimensional, periodic Rayleigh-Benard problem as an example. The main effect investigated here is a strong variation in the density considered primarily as a function of temperature as it is accounted for by the low Mach number equations. Additionally, the effect of temperature-dependent viscosity and heat conductivity is considered. Both types of departure are measured by the nondimensional temperature gradient. They destroy the midplane symmetry of the problem and lead to quantitative and qualitative changes of the flow, in particular in the bifurcation from the conduction state. The present study illustrates how three different approaches-by linear stability analysis, weakly nonlinear analysis, and direct simulation-permit the investigation of complementary aspects of the problem. The first is used to determine the critical Rayleigh number for the onset of convection. The second is well suited for investigating the nature of the bifurcation and shows that the flow in the form of rolls becomes subcritical if the non-Boussinesq character is strong enough. Direct simulations of the flow field confirm this behavior and permit the study of situations far from criticality. It turns out that non-Boussinesq effects are more and more confined to small parts of the domain as the Rayleigh number increases, so that their influence on mean properties of the flow diminishes.
Abstract: We describe a method allowing to compute the perturbed amplitude equation which takes into account geometrical imperfections of the domain of the flow, in a classical problem of nonlinear hydrodynamic instabilities.
Abstract: This paper is devoted to the theoretical study of motions of viscous fluids driven by a constant stress acting on an upper surface of a long rectangular cavity. This problem was originally addressed to the flow behavior of molten metals in open boats driven by thermocapillarity (Bridgman technique solidification). Computations of two-dimensional Navier-Stokes equations show two totally different end circulations. Considering the spatial disturbances of the core flow (e.g., Couette flow), and, using the local theory of bifurcations (center manifold, normal forms), the appearance of two kinds of disturbances corresponding to these end circulations is explained. Moreover, on one hand a condition for the observability of the fully developed Couette flow in terms of aspect ratio (length/height) and Reynolds-Marangoni number is given, and on the other hand the analytical expression of the space periodic flow occurring in the direction of the cold wall is given.
Abstract: We present a synthesis of results obtained from a linear stability analysis of the one-cell Hadley circulation. This flow is observed in the core of the large cavity in horizontal Bridgman configurations. The marginal stability threshold of this motion is given in the case of both rigid horizontal surfaces or upper stress-free surfaces; we also consider either insulating or perfectly conducting boundary conditions. Finally, we look at the influence of thermocapillary forces on the appearance of 2D oscillatory perturbations. This linear analysis allows to describe the competition between 2D and 3D disturbances. Moreover, bifurcated solutions are computed for some boundary conditions by using the differential system on the center manifold. These numerical results allow us to predict, for specified values of the Prandtl number, the form of the solution in the neighbourhood of a neutral stability point. We relate all the results obtained by this way, but only focus on the 2D perturbations in this paper.
Abstract: The main objective of this paper is to show the influence of thermocapillary forces on the buoyancy-driven flow of low-Prandtl-number fluids in shallow cavities with differentially heated endwalls. The horizontal walls are considered as perfectly conducting with a linear temperature profile. The upper horizontal boundary is a free flat surface. Calculations were carried out by solving the Navier-Stokes equations and the computations concern large aspect ratio (length/height) at fixed Prandtl number. For two given Grashof numbers, the Reynolds number was varied from negative to positive values. Such a combined convection problem is also investigated by means of stability theory. We look at the temporal stability of the analytical solution occurring in the core of a large cavity. It is found that critical Grashof number Grc for the onset of the oscillatory regime substantially decreases with small negative Re, and increases with positive Re. But, the thermocapillary forces have again an important stabilizing role for high negative values of Reynolds number. Direct numerical simulations exhibit quite similar behaviour of the evolution of the critical Grashof number Grc, with respect to Re.
Notes: 7TH INTERNATIONAL PHYSICOCHEMICAL HYDRODYNAMICS CONF : GRAVITY EFFECTS IN PCH, CAMBRIDGE, MA, JUN 25-29, 1989
Abstract: The stability of the spherical shape of the free surface of a gas bubble compressed by an incompressible fluid as it appears in the inertial confinement fusion problem is considered. (i) The equations derived by Prosperetti [Accad. Naz. Lincei 62, 196 (1977)] generalizing the Plesset equation are recovered in cases when the outer fluid is nonviscous, the flow being not potential, and it is shown that vorticity may change drastically the results of the potential case, (ii) In the case of viscous external fluid, the equations derived by Prosperetti [Q. Appl. Math. 1, 399 ( 1977)] and other external conditions on a sphere of finite radius are derived. (iii) Assuming that the time scale of the dynamics of the spherical bubble is large with respect to the time scale of the perturbation (frozen assumption), the linear stability of the collapsing bubble is studied numerically. The parameters are here (a) an inertia force (related with acceleration R of the radius of the bubble), (b) the Reynolds number built with the decaying rate of the bubble, (c) surface tension, and (d) the aspect ratio (ratio between the gap width of the viscous fluid and the radius of the bubble). It is shown that the spherical shape is always linearly unstable in the absence of surface tension. In the presence of surface tension, there is a critical inertia parameter value and the most dangerous mode is always stationary. For the case of a large surface tension, the spherical wavenumber l of the most dangerous mode, is low. Finally, it is shown that the Rayleigh-Taylor instability might only be observed for both small aspect ratio and Reynolds number, depending on the surface tension.
Abstract: This paper concerns the onset of oscillatory flows in horizontal layer subject to horizontal temperature gradient. It summarizes the main difficulties encountered, and typical results presented during a recent GAMM Workshop devoted to numerical simulation of oscillatory convection in low-Prandtl-number fluids. Hydrodynamics stability and bifurcation analysis are shown to be useful complementary tools for a better understanding of the onset of oscillations in metallic melts.
Abstract: A particular case of the transition from the Couette flow to the Taylor cells is mathematically studied using the Center Manifold theorem. In this degenerate case, the stability of the Taylor vortex flow is calculated using fifth order terms in amplitude. We describe a method for computing these high order terms. Moreover, the numerical values are obtained with a Fortran program generated by a Macsyma program.
Abstract: This paper is devoted to numerical and theoretical studies of the motions of viscous fluids driven by a constant stress acting in long rectangular enclosures. This problem was originally addressed to study the flow behaviour of molten metals driven by thermocapillarity in open boats during the directional solidification by the Bridgman technique and the study has now been extended to a closely related problem, i.e., the flows occuring in a water-basin under a stress generated by an external wind. We handled this problem by solving the 2D Navier-Stokes equations by an efficient finite-difference technique. Computations have been done for a rectangular basin with various aspect ratios (length/height), and for a large range of the surface Reynolds number. A notable feature of the solutions is the totally different end circulations. Following a previous study of Bye1, we also emphasized this problem through the linear eigenequation resulting from (small) spatial disturbances of a Couette flow solution. We performed this analysis by using an efficient Tau-Chebyshev technique
Abstract: Modelisation and solution of heat and mass transfer problems relevant for material processing are generally hard to be handled, as they often involve 3D unsteady flows, viscous mixtures, phase changes, moving liquid-solid fronts, deforming liquid-gas interfaces, etc.⦠For space applications, material processing benefits of reduced buoyancy convection but can be faced to a strongly increased complexity due to variable g, mainly in manned flight.
Computational techniques used to analyse fluid motions in material processing, accounting for free surface, crystallization front and bulk convection in melt, are reviewed with emphasis to directional crystallization. Hydrodynamics stability and bifurcation analysis are shown to be useful complementary tools for correlating data, and for a better understanding of the physical laws. This last point will be illustrated in the case of the onset of oscillations in metallic melts.
Abstract: The problem of surface tension-driven flows in horizontal liquid layers has been studied experimentally, and theoretically by direct numerical simulation and small perturbation analysis. We focus our attention on situations in which the depth of the fluid (liquid tin; small Prandtl number, Pr=0.015) is small enough to ensure the predominance of the surface tension forces over those due to the buoyancy. The surface velocity has been experimentally obtained for liquid tin layer with various aspect ratio (length to height) in the range 5<A<83. The thermal gradients are ranged from 5 to 40°K/cm. In the numerical study, the Navier-Stokes and energy equations are solved by an efficient finite difference technique. The parameters governing the flow behaviour in the liquid are varied to determine their effects on thermocapillary convection: the Reynolds number 10<Re<2104 and the aspect ratio 2<A<25; with Pr kept constant at Pr=0.015. The linear eigenequation resulting from small spatial disturbances of the Couette flow solution is solved using an Tau-Chebyshev approximation. A notable feature of the theoretical study is the totally different end circulations. In the region near the cold wall a multicell structure is evident. This agrees with the eigensolution which is of complex type, indicating spatial periodicity. In the hot wall region the flow is accelerated to reach the velocity value for the fully-developed Couette flow which is reached under conditions such as Re A<20. The transition from viscous to boundary layer regime occurs for a critical value ( Re A)c of nearly about 200, as deduced from the numerical and experimental results.
Abstract: This paper discusses the flow of a low Prandtl-number liquid contained in a rectangular cavity with differentially heated vertical end walls. It is assumed that the aspect ratio is large enough to impose a horizontal flow in the core. The marginal stability-threshold of this motion is given in two cases, for rigid insulators at both surfaces or for upper stress-free insulator surface. First bifurcated solutions are computed, using the differential system on the center manifold. The numerical results make it possible to predict, for specified values of the Prandtl number, the form of the solution. In the rigid-rigid case either study horizontal cells or periodic traveling waves are obtained. In the rigid-free case, two forms of periodic solutions are obtained, either traveling waves or stationary waves.
Abstract: This book provides the fundamental basics for solving fluid structure interaction problems, and describes different algorithms and numerical methods used to solve problems where fluid and structure can be weakly or strongly coupled. These approaches are illustrated with examples arising from industrial or academic applications. Each of these approaches has its own performance and limitations. The added mass technique is described first. Following this, for general coupling problems involving large deformation of the structure, the Navier-Stokes equations need to be solved in a moving mesh using an ALE formulation. The main aspects of the fluid structure coupling are then developed. The first and by far simplest coupling method is explicit partitioned coupling. In order to preserve the flexibility and modularity that are inherent in the partitioned coupling, we also describe the implicit partitioned coupling using an iterative process. In order to reduce computational time for large-scale problems, an introduction to the Proper Orthogonal Decomposition (POD) technique applied to FSI problems is also presented. To extend the application of coupling problems, mathematical descriptions and numerical simulations of multiphase problems using level set techniques for interface tracking are presented and illustrated using specific coupling problems. Given the bookâs comprehensive coverage, engineers, graduate students and researchers involved in the simulation of practical fluid structure interaction problems will find this book extremely useful.
Abstract: Spatiotemporal intermittency manifests itself by the coexistence of laminar and turbulent domains for the same value of the control parameter. In the Taylor- Dean system, the distributions of laminar domains size after algebraic and exponential regimes allow for a determination of critical properties in an analogy with directed percolation. In the Couette-Taylor system, only algebraic distribution of laminar domains size has been evidenced. A turbulent spiral coexists with laminar spiral destroying the occurrence of exponential regime.
Abstract: The polymer coextrusion process is a new method of sheet metal lining. It allows to substitute lacquers for steel protection in food packaging industry. The coextrusion process may exhibit flow instabilities at the interface between the two polymer layers. The objective of this study is to check the influence of processing and rheology parameters on the instabilities. Finite elements numerical simulations of the coextrusion allow to investigate various stable and instable flow configurations.
Notes: 10th ESAFORM Conference on Material Forming, Zaragoza, SPAIN, APR 18-20, 2007
Abstract: This workâs context is an industrial project aiming the accurate modeling of the injection molding process [1]. 3D numerical simulation of the different stages is considered: during processing, anisotropy of the stress state build up affects its mechanical, optical or dimensional properties, and induces warpage once the part is ejected. A first example of injection molding of reinforced thermoplastics will be treated. In this case, we will consider that during the injection step, an orientation will be induced by the flow. Furthermore, the thermoplastic matrix will pass from the liquid to the solid state, and orientation and stresses will remain frozen. Evolution of orientation or extra stress is computed using the Folgar and Tucker equation, with continuous or discontinuous approximations. Results are obtained in a 3D complex industrial part.
Notes: 9th International Conference on Numerical Methods in Industrial Forming Processes (NUMIFORM 07), Oporto, PORTUGAL, JUN 17-21, 2007
Abstract: The aim of this paper is to propose a multi-domain approach to model the flow motion of concrete flow. We mainly deal with free surfaces, material interfaces and fiber orientation. The main application behind this paper is related with the addition of rigid long metal pieces to fresh concrete in order to improve the final mechanical properties of concrete beams. Binghamâs rheology is used for fresh concrete behavior, whereas the fiber orientation is described by Folgar and Tucker equation. The velocity field is computed using a classical finite element method whereas the evolution of orientation tensors is solved by a space-time discontinuous Galerkin method. Furthermore, a specific method has been developed in order to compute accurately the moving free surface.
Notes: 10th ESAFORM Conference on Material Forming, Zaragoza, SPAIN, APR 18-20, 2007
Abstract: Fibre-like particles into a polymer matrix enables to enhance the mechanical properties of a composite material. The degree of enhancement depends strongly on the fibre orientation which depends itself on the required flow during the forming process in moulding for instance. The numerical modelling of a fluid with fibres deals with an evolution equation involving the second orientation tensor. However, it results from homogenisation procedures from Jeffery equation for the orientation of a single particle and from Folker-Plank equation for the probability distribution of orientation. Therefore, this approach has a limited domain of validity, depending on the shape, the aspect ratio and the volume fraction of fibres. We propose here to simulate directly the motion of a dense population of fibers in a polymeric fluid, taking into account the exact particle interaction, by using a multidomain approach in a global Finite Element calculation. The first interest of this direct approach is to avoid the need of an explicit form of drag and lubrication forces acting between fibres. This presentation will focus on the influence of the particle shape on the motion of a single particle (Jefferyâs equation works well provided that an equivalent aspect ratio is used), the fibre motion near a wall is still described if an increased effective shear rate is used, the numerical calculation with a great number of fibres and the averaging to produce macroscopic properties of fibre suspensions. In this way, we can determine the evolution of Folgar-Tucker diffusion constant and the validity of closure approximation with respect to the volume fraction of fibres.
Notes: 8th International Conference on Numerical Methods in Industrial Forming Processes, Columbus, OH, JUN 13-17, 2004
Abstract: Spatiotemporal intermittency manifests itself by the coexistence of laminar and turbulent domains for the same value of the control parameter. In the Taylor-Dean system, the distributions of laminar domains size after algebraic and exponential regimes allow for a determination of critical properties in an analogy with directed percolation. In the Couette-Taylor system, only algebraic distribution of laminar domains size has been evidenced. A turbulent spiral coexists with laminar spiral destroying the occurrence of exponential regime.
Notes: 11th International Couette-Taylor Workshop, Bremen, Germany, JUL 20-23, 1999
