# C07Vorticity concentration for fluid simulation

## Development of new algorithms for fluid simulation, visualization and processing.

Fluid simulations for Computer Graphics create challenges which are outside the scope of standard techniques in Computational Fluid Dynamics. One of the main problems is that large scale phenomena are often driven by structures like vortex filaments or vortex sheets which are thin in compare to feasible grid resolutions and quickly destroyed by numerical diffusion. In this project we develop new tools for fluid simulation, visualization and processing based on a quantum mechanical description of incompressible fluids, which is capable to reproduce phenomena driven by thin vortical structures faithfully even on coarse grids.

#### Scientific Details+

The project is concerned with the development of new algorithms for fluid simulation, visualization and processing based on a quantum mechanical description of incompressible fluids. Here the state of the fluid is described by a $C^2$-valued wave function evolving by the Schrödinger equation subject to incompressibility constraints. The dynamical system underlying this 'incompressible Schrödinger flow' (ISF) is a Hamiltonian system with Hamiltonian consisting of the kinetic energy of the fluid and an additional energy of Landau-Lifschitz type, which ensures that the dynamics due to thin vortical structures are faithfully reproduced.

We want to investigate the properties of ISF as a Hamiltonian system. There are indications that various limits (like infinitely sharp vortex sheets or filaments) are the same as the ones that arise from the coadjoint orbit picture (Kostant-Kirillov construction) for the group of orientation-preserving diffeomorphisms.

We also aim at a deeper understanding of the connection of ISF and the Landau-Lifshitz equation - an equation which describes spin waves and also is known for exhibiting vortex ring dynamics.

Moreover it would be interesting to study and explore the practical usefulness of symplectic integrators in our context.

For artists it is of practical importance to have control over the flow. We will develop tools that allow artists to design initial conditions that are suitable for their intentions. This is closely related to the construction of a wave function representing a given velocity field. On grids of higher resolution this becomes a challenging task and will need the development of adapted numerical tools. Moreover, we want to find ways to incorporate external forces (like buoyancy) or moving obstacles into our equations.

The representation by a wave function offers new ways to visualize and process fluids. We are going to explore this.

There are also indications that, using $C^n$-valued wave functions, it might be possible to model vorticity concentration on two-dimensional sheets that have not yet rolled up into filaments. We want to achieve an evolution of vorticity into sharp sheets and filaments gradually over time, allowing for a smooth transition.

What else can be done switching to wave functions with more than two components? Can we model multi-phase fluids? Are there other interesting physics to be modelled in this way?

#### Publications+

##### Papers
###### Inside Fluids: Clebsch Maps for Visualization and Processing

Authors: Chern, Albert and Knöppel, Felix and Pinkall, Ulrich and​ ​ Schröder, Peter
Journal: ACM Trans. Graph., 36(4):142:1--142:11
Date: Jul 2017
DOI: 10.1145/3072959.3073591
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###### Hierarchical Vorticity Skeletons

Authors: Eberhardt, Sebastian and Weissmann, Steffen and Pinkall, Ulrich and Thuerey, Nils
Journal: Proceedings of the Symposium on Computer Animation (SCA '12), to appear:11
Date: Jun 2017
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###### Schrödinger's Smoke

Authors: Chern, Albert and Knöppel, Felix and Pinkall, Ulrich and​ ​ Schröder, Peter and Weißmann, Steffen
Journal: ACM Trans. Graph., 35:77:1--77:13
Date: Jul 2016
DOI: 10.1145/2897824.2925868
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###### Complex Line Bundles over Simplicial Complexes and their Applications

Authors: Knöppel, F. and Pinkall, U.
In Collection: Advances in Discrete Differential Geometry, Springer
Date: 2016
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###### Close-to-conformal deformations of volumes

Authors: Chern, A. and Pinkall, U. and Schröder, P.
Journal: ACM Transactions on Graphics, 34(4):56
Date: 2015

###### Filament-based smoke with vortex shedding and variational reconnection

Authors: Weißmann, S. and Pinkall, U.
Journal: ACM Transactions on Graphics, 29
Date: 2010

##### PhD thesis
###### Complex Line Bundles over Simplicial Complexes

Author: Knöppel, F.
Date: 2015
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#### Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin
E-Mail: pinkall[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~pinkall/

#### Dr. Albert Chern   +

Projects: A05, C07
University: TU Berlin
E-Mail: chern[at]math.tu-berlin.de

#### Dr. Felix Knöppel   +

Projects: A01, C07
University: TU Berlin
E-Mail: knoeppel[at]math.tu-berlin.de

#### Christoph Seidel   +

Projects: C07
University: TU Berlin
E-Mail: seidel[at]math.tu-berlin.de