## Berichte des Fraunhofer-Instituts für Techno- und Wirtschaftsmathematik (ITWM Report)

- 197
- Fluid-fiber-interactions in rotational spinning process of glass wool production (2010)
- 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.

- 192
- Asymptotic transition from Cosserat rod to string models for curved viscous inertial jets (2010)
- 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.

- 186
- On adjoint-based optimization of a free surface Stokes flow (2010)
- This work deals with the optimal control of a free surface Stokes flow which responds to an applied outer pressure. Typical applications are fiber spinning or thin film manufacturing. We present and discuss two adjoint-based optimization approaches that differ in the treatment of the free boundary as either state or control variable. In both cases the free boundary is modeled as the graph of a function. The PDE-constrained optimization problems are numerically solved by the BFGS method, where the gradient of the reduced cost function is expressed in terms of adjoint variables. Numerical results for both strategies are finally compared with respect to accuracy and efficiency.

- 172
- Modeling and validation of a stochastic drag for fibers in turbulent flows (2009)
- 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.

- 167
- Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes (2009)
- 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].

- 156
- Hierarchy of mathematical models for production processes of technical textiles (2009)
- 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.

- 145
- Adjoint based optimal control using mesh-less discretizations (2008)
- An easy numerical handling of time-dependent problems with complicated geometries, free moving boundaries and interfaces, or oscillating solutions is of great importance for many applications, e.g., in fluid dynamics (free surface and multiphase flows, fluid-structure interactions [22, 18, 24]), failure mechanics (crack growth and propagation [4]), magnetohydrodynamics (accretion disks, jets and cloud simulation [6]), biophysics and -chemistry. Appropriate discretizations, so-called mesh-less methods, have been developed during the last decades to meet these challenging demands and to relieve the burden of remeshing and successive mesh generation being faced by the conventional mesh-based methods, [16, 10, 3]. The prearranged mesh is an artificial constraint to ensure compatibility of the mesh-based interpolant schemes, that often conflicts with the real physical conditions of the continuum model. Then, remeshing becomes inevitable, which is not only extremely time- and storage consuming but also the source for numerical errors and hence the gradual loss of computational accuracy. Apart from this advantage, mesh-less methods also lead to fundamentally better approximations regarding aspects, such as smoothness, nonlocal interpolation character, flexible connectivity, refinement and enrichment procedures, [16]. The common idea of mesh-less methods is the discretization of the domain of interest by a finite set of independent, randomly distributed particles moving with a characteristic velocity of the problem. Location and distribution of the particles then account for the time-dependent description of the geometry, data and solution. Thereby, the global solution is linearly superposed from the local information carried by the particles. In classical particle methods [20, 21], the respective weight functions are Dirac distributions which yield solutions in a distributional sense.

- 115
- Dynamics of curved viscous fibers with surface tension (2007)
- 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.

- 112
- Hydrodynamic limit of the Fokker-Planck-equation describing fiber lay-down processes (2007)
- 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.

- 102
- A Stochastic Model for the Fiber Lay-down Process in the Nonwoven Production (2006)
- 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.