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Publication# Numerical simulation of orbitally shaken viscous fluids with free surface

Marco Discacciati, David Hacker, Alfio Quarteroni, Samuel Quinodoz, Stéphanie Tissot, Florian Maria Wurm

*Wiley-Blackwell, *2013

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Résumé

Orbitally shaken bioreactors are an emerging alternative to stirred-tank bioreactors for large-scale mammalian cell culture, but their fluid dynamics is still not well defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors. To simulate the fluid behavior and the free surface shape, we developed a numerical method based on the finite element framework. We found that the large density ratio between the liquid and the gas phases induced unphysical results for the free surface shape. We therefore devised a new pressure correction scheme to deal with large density ratios. The simulations operated with this new scheme gave values of wave amplitude similar to the ones measured experimentally. These simulations were used to calculate the shear stress and to study the mixing principle in orbitally shaken bioreactors. Copyright (C) 2012 John Wiley & Sons, Ltd.

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A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999Mammalian cell cultures have become a major topic of research in the biopharmaceutical industry. This kind of cells requires specific conditions to grow. In this thesis, we study the hydrodynamics of orbitally shaken reactors (OSR), a recently introduced kind of bioreactors for mammalian cell cultures that represents a simple to operate and cheap alternative to commonly used reactors such as stirred tanks. OSRs can provide suitable conditions for small scale cell cultures, however a deeper understanding of the principles governing the OSRs is required to exploit their full potentiality and proceed with scaling up. This work aims at shedding light into the mechanisms of the OSRs through computational fluid dynamics. OSRs are only partially filled with liquid medium, the remaining space is occupied by air. When an OSR is agitated, the interface between the two phases moves and creates different shapes. This interface is at the heart of the simulation of OSRs: not only its location is part of the problem, but it can also carry singularities. In particular, the pressure has usually a low regularity in the vicinity of the interface and numerical methods might underperform if the singularity is not treated in an appropriate manner. This motivated the study of an elliptic problem in a medium with an internal interface carrying discontinuities. In this work, we devise a novel method called SESIC to solve this kind of problem. It uses the a priori knowledge to improve the numerical accuracy in the vicinity of the interface by removing the singularities. We prove that this method yields optimal orders of convergence in H1 and L2 norm. Numerical tests also show that optimal orders can be obtained in the L∞ norm in some cases. Regularized integration is also investigated with the perspective of further simplifying the scheme. It is found that, if the regularization bandwidth is suitably defined, good approximations can still be obtained, even if the convergence rate is decreased. We apply then the methodology of the SESIC method to the approximation of the two-phase Navier-Stokes equations, which amounts to correct the pressure. If adapted integration is used with it, the density and viscosity can be kept discontinuous across the interface without creating spurious velocity, as shown by numerical experiments. The sharp treatment of the discontinuities improves the accuracy of the simulations by retaining the physical meaning of the phenomena independently of the mesh size. We also pay attention to the boundary conditions used, which must be suitably chosen to allow the interface motion but still reproduce the wall friction. We show that imposing the zero normal component of the velocity yields the best results for the no-penetration condition and that it must be employed with a correction term to avoid spurious velocities. Robin-type conditions are used for the tangential components to recover the no-slip condition far from the contact line. Specific tests are performed to assess the quality of the different components of the method. We also compare it with a regularized density/viscosity method and show that the sharp treatment of these physical quantities improves the quality of the simulation. The scalability properties of the method are also investigated and the bottlenecks pointed out. Our method is then validated in various ways with experimental data. First of all, glycerine filled OSRs are simulated and we show that our method reproduces accurately the amplitude of the generated wave. The sensibility of the results with respect to the Robin condition is shown to be weak. We investigate then water filled OSRs. The different wave patterns, either breaking or non-breaking, single or multiple, observed experimentally are reproduced for various configurations. In particular, triple waves are obtained as well. We use laser Doppler velocimetry measures of the velocity field to further validate our simulations. The hydrodynamic stress and the mixing pattern of the different regimes are evaluated and put into relationship with the wave shape. Finally, we investigate the modelling of the cell culture by devising a system non-linear ODEs which represents the evolution in time of the main nutriments and wastes and the growth of the cell population. The behaviour of the cell culture is well reproduced, but some phenomena remain unexplained. In particular, our model contains a toxic waste whose actual identity is discussed. Two alternative scenarios are proposed to improve the model.

The subject of this thesis is the numerical simulation of viscous free-surface flows in naval engineering applications. State-of-the-art numerical methods based on the solution of the Navier-Stokes equations are used to predict the flow around different classes of boats. We investigate the role of the Computational Fluid Dynamics in the design of racing boats, such as America's Cup yachts and Olympic class rowing hull. The mathematical models describing the different aspects of the physical problem, as well as the numerical methods adopted for their solution, are introduced and critically discussed. The different phases of the overall numerical simulation procedure, from grid generation through the solution of the flow equations to the post-processing of the results, are described. We present the numerical simulations that have been performed to investigate the role of different design parameters in the conception of America's Cup yachts and we describe how the results obtained from the simulations are integrated into the overall design process. The free-surface flow around an Olympic rowing boat is also considered. We propose a simplified approach to take into account the effect of the boat dynamics in the prediction of the hydrodynamic forces acting on the boat. Based on the results of the simulations, we propose a new design concept and we investigate its potential benefits on the boat performances. One of the aspects that is found to be not completely satisfactory, within the standard numerical methods adopted, is the modelling of complex free-surface flows. The second part of this thesis is devoted to a more theoretical and methodological investigation of this aspect. In particular, we present and analyse a new numerical method based on the level set approach for the solution of two-fluid flows. The numerical scheme based on a finite element discretization is introduced and different critical aspects of its implementation are discussed. In particular, we present and analyse a new technique for the stabilization of the advection equation associated to the level set problem. Moreover, we propose a new reinitialization procedure for the level set function which plays a crucial role in the accuracy of the algorithm. The convergence properties of this procedure are analysed and comparisons with more standard approaches are presented. Finally, the proposed method has been used to solve a variety of test cases concerning time dependent two-fluid viscous flows. The results of the simulation are presented and discussed.