CE40 - Mathématiques, informatique théorique, automatique et traitement du signal

# Singular flows: boundary layers, vortex filaments, wave-structure interaction – SingFlows

## SingFlows

Singular Flows: boundary layers, vortex filaments, wave-structure interaction.

### Objectives of the project

The project SingFlows aims at a better understanding of three topics in fluid dynamics: <br />i) The description of anisotropic flows, like boundary layers, shallow water or pipe flows. <br />ii) The description of vortex dynamics in slightly viscous fluids.<br />iii) The interaction between water waves and fixed or floating structures.<br />These topics originate from different contexts, but they have strong connections at the mathematical level. In all three settings, the flows can be described by Euler or Navier-Stokes type equations, and the associated solutions exhibit spatial singularities. For instance, the contact line between the water surface and a floating object is no more than Lipschitz. Vortex filaments in ideal fluids are modeled by Dirac masses along curves. In the Navier-Stokes evolution, such filaments are instantaneously regularised at positive times, but they still exhibit a singular dependence in the Reynolds number as it becomes large, similarly to what happens for boundary layers. From the numerical point of view, the common difficulty is that these singularities cannot be captured by direct computations, as they require too many grid points. It is then necessary to derive reduced models, that allow to retain the main features of the singular region, and/or its effect on the large scale flow. The goal of the SingFlows project is to develop innovative mathematical methods, as well as efficient<br />reduced models and codes, to improve the description of the singular flows involved in i)-ii)-iii).

1. To develop new mathematical methods for fluid models at low regularity.
2. To obtain effective reduced models for singular flows.
3. To develop efficient numerical codes based on these reduced models.

Result 1. Beyond Prandtl : understanding the stability of interactive boundary layer models.

Result 2. Improvement of friction modelling in anisotropic flows

Result 3. Description of vortex interaction in slightly viscous fluids
- Single vortex filament
- Interaction of several vortex filaments
- Interaction with material boundaries
- Interaction with shear

Result 4. Understanding singularity formation on vortex structures via reduced models
- Vortex filaments with corners
- Interaction between several vortices

Result 5. Modeling and theory of wave-structure interactions
- Theoretical understanding of the contact line.
- Regularizing effects of viscosity

Result 6. Understanding related models
- The wave-maker problem: wave generation
- The wave-maker problem: artificial boundary conditions and wave absorption
- Connection to congested flow models

We hope to be able to give a rigorous justification to the use of interactive boundary layer and triple deck theories. In this way, we will bridge a gap between the mathematical community and the mechanics/engineering community where these theories have been used success- fully - without much justification.

We plan to obtain an improved modelling of friction in shallow water or pipe flows, starting from the Navier-Stokes equations. This will provide a significant improvement compared to most derivations, where the friction terms are given a priori, mostly through empirical laws. A very stimulating perspective, probably more in the long term, is to be able to derive a simple dimensional friction law, involving the water height and the mean velocity. This law will then be included in the usual Saint-Venant system. A breakthrough would be to recover in this way the usual Chézy’s or Manning’s empirical laws.

As regards the dynamics of vortices, we expect significant progress in at least two directions: towards a rigorous justification of the binormal flow, and towards a mathematical description of the interactions between vortices and material walls.

Concerning the mathematical analysis of wave-structure model, the long term hope is a full mathe- matical understanding of the system. In our efforts towards this goal, we will have to solve open problems of independent interest on compressible-incompressible models - that arise in other situations involving congested flows such as granular media, traffic jams, social hydrodynamics - and on mixed initial-boundary value problems for dispersive perturbations of hyperbolic systems - very important for many applications in hydrodynamics.

Beyond improving our theoretical knowledge, one ambition of the SingFlows project is to be of an applied nature, and our activities around the theme of wave-structure interactions are very promising in this respect. These activities benefit from the interaction with european specialists in the domain of wave energy conversion, via a collaboration with the consortium of the MIDWEST OceanERANET project. We have furthermore initiated another collaboration with the technology transfer agency Tecnalia, in order to identify the problems that are most relevant for practical applications.

The objective of SingFlows is to develop mathematical and numerical tools for the analysis of three problems in fluid dynamics: the behaviour of anisotropic flows (boundary layers, shallow water flows), the dynamics of vortical structures, and the evolution of fixed or floating structures in water waves. Our will to unify these different problems is natural, because they share many mathematical features.

The underlying keypoint is that they are described by singular solutions of Euler or Navier-Stokes equations. The word singular refers here:
- either to a lack of smoothness: it applies for instance to vortex filaments, which are Dirac masses along curves, or to the contact line between water and the floating structure,
- or to a singular dependence of the solution with respect to a parameter, typically the Reynolds number (like in boundary layers).
The connection between the two points of view is usually made by viscous regularization of the non-smooth structure, or conversely by taking the vanishing limit of the parameter.

More generally, the three problems considered in SingFlows involve flows with very small scales. Therefore a relevant description requires the derivation of reduced models.

Beyond these common mathematical challenges, which are at the core of the project, the problems studied in SingFlows are intrinsically of an applied nature. They have concrete implications (for river flows, blood circulation, the wear of floating structures), and any quantitative understanding requires numerical simulations, as well as a knowledge of realistic settings. This is why the SingFlows project is based on an integrated team of 25 people with expertise in partial differential equations, numerical analysis, and in the computation and physics of fluid dynamics. This team has pre-existing connections, which will make the task implementation easier. This task implementation follows a careful schedule, with identification of short-, mid- and long-term objectives. It includes efforts towards the dissemination of our results, both to international specialists and general audience.

Several breakthrough results are expected:
- an improved description of friction laws in shallow water flows.
- the justification of the binormal flow approximation through a vanishing viscosity limit.
- the development of robust and efficient numerical codes for wave-structure interaction.

## Project coordinator

Monsieur David Gerard-Varet (Institut de mathématiques de Jussieu - Paris Rive Gauche)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

IF Institut Fourier
IMB Institut de mathématiques de Bordeaux
IMJ-PRG Institut de mathématiques de Jussieu - Paris Rive Gauche

Help of the ANR 263,628 euros
Beginning and duration of the scientific project: January 2019 - 48 Months