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Mon, 24 Jan 2011 08:55:26 +0100Mon, 24 Jan 2011 08:55:26 +0100A 3d model for fiber lay-down in nonwoven production processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2277
In this paper a three dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials is derived. The model is based on stochastic diferential equations describing the resulting position of the fiber on the belt under the influence of turbulent air ows. The model presented here is an extension of an existing surrogate model, see [6, 3].A. Klar; J. Maringer; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2277Mon, 24 Jan 2011 08:55:26 +0100Fluid-fiber-interactions in rotational spinning process of glass wool production
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2276
The optimal design of rotational production processes for glass wool manufacturing poses severe computational challenges to mathematicians, natural scientists and engineers. In this paper we focus exclusively on the spinning regime where thousands of viscous thermal glass jets are formed by fast air streams. Homogeneity and slenderness of the spun fibers are the quality features of the final fabric. Their prediction requires the computation of the fuidber-interactions which involves the solving of a complex three-dimensional multiphase problem with appropriate interface conditions. But this is practically impossible due to the needed high resolution and adaptive grid refinement. Therefore, we propose an asymptotic coupling concept. Treating the glass jets as viscous thermal Cosserat rods, we tackle the multiscale problem by help of momentum (drag) and heat exchange models that are derived on basis of slender-body theory and homogenization. A weak iterative coupling algorithm that is based on the combination of commercial software and self-implemented code for ow and rod solvers, respectively, makes then the simulation of the industrial process possible. For the boundary value problem of the rod we particularly suggest an adapted collocation-continuation method. Consequently, this work establishes a promising basis for future optimization strategies.W. Arne; N. Marheineke; J. Schnebele; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2276Mon, 24 Jan 2011 08:54:38 +0100Asymptotic transition from Cosserat rod to string models for curved viscous inertial jets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2226
This work deals with the modeling and simulation of slender viscous jets exposed to gravity and rotation, as they occur in rotational spinning processes. In terms of slender-body theory we show the asymptotic reduction of a viscous Cosserat rod to a string system for vanishing slenderness parameter. We propose two string models, i.e. inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. We investigate the existence regimes of the string models in the four-parametric space of Froude, Rossby, Reynolds numbers and jet length. The convergence regimes where the respective string solution is the asymptotic limit to the rod turn out to be disjoint and to cover nearly the whole parameter space. We explore the transition hyperplane and derive analytically low and high Reynolds number limits. Numerical studies of the stationary jet behavior for different parameter ranges complete the work.W. Arne; N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2226Tue, 10 Aug 2010 09:51:57 +0200Modeling and validation of a stochastic drag for fibers in turbulent flows
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2150
The understanding of the motion of long slender elastic fibers in turbulent flows is of great interest to research, development and production in technical textiles manufacturing. The fiber dynamics depend on the drag forces that are imposed on the fiber by the fluid. Their computation requires in principle a coupling of fiber and flow with no-slip interface conditions. However, theneeded high resolution and adaptive grid refinement make the direct numerical simulation of the three-dimensional fluid-solid-problem for slender fibers and turbulent flows not only extremely costly and complex, but also still impossible for practically relevant applications. Embedded in a slender body theory, an aerodynamic force concept for a general drag model was therefore derived on basis of a stochastic k-o; description for a turbulent flow field in [23]. The turbulence effects on the fiber dynamics were modeled by a correlated random Gaussian force and its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow-dependent amplitude. The concept was numerically studied under the conditions of a melt-spinning process for nonwoven materials in [24] – for the specific choice of a non-linear Taylor drag model. Taylor [35] suggested the heuristic model for high Reynolds number flows, Re in [20, 3 · 105], around inclined slender objects under an angle of attack of alpha in (pi/36, pi/2] between flow and object tangent. Since the Reynolds number is considered with respect to the relative velocity between flow and fiber, the numerical results lackaccuracy evidently for small Re that occur in cases of flexible light fibers moving occasionally with the flow velocity. In such a regime (Re << 1), linear Stokes drag forces were successfully applied for the prediction of small particles immersed in turbulent flows, see e.g. [25, 26, 32, 39], a modifiedStokes force taking also into account the particle oscillations was presented in [14]. The linear drag relation was also conferred to longer filaments by imposing free-draining assumptions [29, 8]. Apart from this, the Taylor drag suffers from its non-applicability to tangential incident flow situations (alpha = 0) that often occur in fiber and nonwoven production processes.N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2150Mon, 30 Nov 2009 09:33:50 +0100Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2141
The rotational spinning of viscous jets is of interest in many industrial applications, including pellet manufacturing [4, 14, 19, 20] and drawing, tapering and spinning of glass and polymer fibers [8, 12, 13], see also [15, 21] and references within. In [12] an asymptotic model for the dynamics of curved viscous inertial fiber jets emerging from a rotating orifice under surface tension and gravity was deduced from the three-dimensional free boundary value problem given by the incompressible Navier-Stokes equations for a Newtonian fluid. In the terminology of [1], it is a string model consisting of balance equations for mass and linear momentum. Accounting for inner viscous transport, surface tension and placing no restrictions on either the motion or the shape of the jet’s center-line, it generalizes the previously developed string models for straight [3, 5, 6] and curved center-lines [4, 13, 19]. Moreover, the numerical results investigating the effects of viscosity, surface tension, gravity and rotation on the jet behavior coincide well with the experiments of Wong et.al. [20].W. Arne; N. Marheineke; A. Meister; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2141Fri, 16 Oct 2009 08:57:18 +0200Hierarchy of mathematical models for production processes of technical textiles
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2084
In this work we establish a hierarchy of mathematical models for the numerical simulation of the production process of technical textiles. The models range from highly complex three-dimensional fluid-solid interactions to one-dimensional fiber dynamics with stochastic aerodynamic drag and further to efficiently handable stochastic surrogate models for fiber lay-down. They are theoretically and numerically analyzed and coupled via asymptotic analysis, similarity estimates and parameter identification. Themodel hierarchy is applicable to a wide range of industrially relevant production processes and enables the optimization, control and design of technical textiles.A. Klar; N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2084Wed, 13 May 2009 13:37:30 +0200An energy conserving numerical scheme for the dynamics of hyperelastic rods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2075
A numerical method for special Cosserat rods based on Antman’s description [1] is developed for hyperelastic materials and potential forces. This method preserves the relevant properties of the underlying PDE system, namely the orthonormality of the directors and the conservation of the energy.Th. Fütterer; A. Klar; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2075Wed, 25 Mar 2009 14:47:53 +0100Dynamics of curved viscous fibers with surface tension
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1969
In this paper we extend the slender body theory for the dynamics of a curved inertial viscous Newtonian fiber [23] by the inclusion of surface tension in the systematic asymptotic framework and the deduction of boundary conditions for the free fiber end, as it occurs in rotational spinning processes of glass fibers. The fiber ow is described by a three-dimensional free boundary value problem in terms of instationary incompressible Navier-Stokes equations under the neglect of temperature dependence. From standard regular expansion techniques in powers of the slenderness parameter we derive asymptotically leading-order balance laws for mass and momentum combining the inner viscous transport with unrestricted motion and shape of the fiber center-line which becomes important in the practical application. For the numerical investigation of the effects due to surface tension, viscosity, gravity and rotation on the fiber behavior we apply a fnite volume method with implicit flux discretization.N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1969Wed, 28 May 2008 10:23:24 +0200Hydrodynamic limit of the Fokker-Planck-equation describing fiber lay-down processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1966
In this paper, a stochastic model [5] for the turbulent fiber laydown in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corresponding to large noise values, the transient and stationary joint probability distributions are determined using the method of multiple scales and the Chapman-Enskog method. Moreover, exponential convergence towards the stationary solution is proven for the reduced problem. For special choices of the industrial parameters, the stochastic limit process is an Ornstein{Uhlenbeck. It is a good approximation of the fiber motion even for moderate noise values. Moreover, as shown by Monte{Carlo simulations, the limiting process can be used to assess the quality of nonwoven materials in the industrial application by determining distributions of functionals of the process.L. Bonilla; T. Götz; A. Klar; N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1966Wed, 28 May 2008 10:23:04 +0200Numerical evidance for the non-existing of solutions of the equations desribing rotational fiber spinning
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1961
Abstract. The stationary, isothermal rotational spinning process of fibers is considered. The investigations are concerned with the case of large Reynolds (± = 3/Re ¿ 1) and small Rossby numbers (\\\" ¿ 1). Modelling the fibers as a Newtonian fluid and applying slender body approximations, the process is described by a two–point boundary value problem of ODEs. The involved quantities are the coordinates of the fiber’s centerline, the fluid velocity and viscous stress. The inviscid case ± = 0 is discussed as a reference case. For the viscous case ± > 0 numerical simulations are carried out. Transfering some properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for ± > 3\\\"2 no physical relevant solution can exist. A possible interpretation of the above coupling of ± and \\\" related to the die–swell phenomenon is given.Th. Götz; A. Klar; A. Unterreiter; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1961Wed, 28 May 2008 10:22:37 +0200A Stochastic Model for the Fiber Lay-down Process in the Nonwoven Production
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1816
In this paper we present and investigate a stochastic model for the lay-down of fibers on a conveyor belt in the production process of nonwovens. The model is based on a stochastic differential equation taking into account the motion of the ber under the influence of turbulence. A reformulation as a stochastic Hamiltonian system and an application of the stochastic averaging theorem lead to further simplications of the model. Finally, the model is used to compute the distribution of functionals of the process that might be helpful for the quality assessment of industrial fabrics.T. Götz; A. Klar; N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1816Tue, 28 Nov 2006 19:19:26 +0100A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1742
We present the application of a meshfree method for simulations of interaction between fluids and flexible structures. As a flexible structure we consider a sheet of paper. In a two-dimensional framework this sheet can be modeled as curve by the dynamical Kirchhoff-Love theory. The external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet. This pressure difference is computed using the Finite Pointset Method (FPM) for the incompressible Navier-Stokes equations. FPM is a meshfree, Lagrangian particle method. The dynamics of the sheet are computed by a finite difference method. We show the suitability of the meshfree method for simulations of fluid-structure interaction in several applications.S. Tiwari; S. Antonov; D. Hietel; J. Kuhnert; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1742Wed, 31 May 2006 14:02:33 +0200Slender Body Theory for the Dynamics of Curved Viscous Fibers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1713
The paper at hand presents a slender body theory for the dynamics of a curved inertial viscous Newtonian ber. Neglecting surface tension and temperature dependence, the ber ow is modeled as a three-dimensional free boundary value problem via instationary incompressible Navier-Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter leading-order balance laws for mass (cross-section) and momentum are derived that combine the unrestricted motion of the ber center-line with the inner viscous transport. The physically reasonable form of the one-dimensional ber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms.S. Panda; R. Wegener; N. Marheinekereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1713Wed, 08 Mar 2006 18:34:11 +0100Fiber Dynamics in Turbulent Flows -Part I: General Modeling Framework -Part II: Specific Taylor Drag
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1665
Part I: General Modeling Framework The paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber. Independent of the choice of the drag model, a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow. Its construction as centered differentiable Gaussian field complies thereby with the requirements of the stochastic k-turbulence model and Kolmogorov’s universal equilibrium theory on local isotropy. Part II: Specific Taylor Drag In [12], an aerodynamic force concept for a general air drag model is derived on top of a stochastic k-epsilon description for a turbulent flow field. The turbulence effects on the dynamics of a long slender elastic fiber are particularly modeled by a correlated random Gaussian force and in its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow - dependent amplitude. The paper at hand now presents quantitative similarity estimates and numerical comparisons for the concrete choice of a Taylor drag model in a given application.N. Marheineke; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1665Tue, 13 Sep 2005 12:48:57 +0200Multivalued Fundamental Diagrams And Stop And Go Waves For Continuum Traffic Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1488
In the present paper a kinetic model for vehicular traffic leading to multivalued fundamental diagrams is developed and investigated in detail. For this model phase transitions can appear depending on the local density and velocity of the flow. A derivation of associated macroscopic traffic equations from the kinetic equation is given. Moreover, numerical experiments show the appearance of stop and go waves for highway traffic with a bottleneck.M. Günther; A. Klar; T. Materne; R. Wegenerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1488Mon, 02 Feb 2004 15:32:08 +0100