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## Natural boundary conditions involving pressure drop, without energy conservation

Many practical problems in fluid dynamics are studied and conceptualized in unbounded domains. Then, these domains have to be truncated to allow the computation of the flow field in a finite computational domain. As a consequence, boundary conditions associated with these artificial boundaries are to be defined.

Navier–Stokes equations: applications to air and blood flows. In this case, one can simply decide to keep the essential boundary condition at the inlet and leave the solution and the test space free at the outlet. This method is common: actually, omitting the term R (pn ru n) v in the variational formulation, we are imposing a zero-normal-stress condition at the outflow of the domain where the velocity is not known [62, 110]. This kind of boundary condition is considered in [70] and amounts to impose pn ru n = 0 on . Then we have a homogeneous Neumann condition that occurs naturally in the variational formulation on all boundaries where no condition is imposed on the velocity. These boundary conditions are called free outflow boundary condition (see [70]) since they are commonly used as passive conditions at the artificial boundaries. The variational problems are the same that Problem P2.2.1 and Problem P2.2.2 without the boundary integral on , ([u; p]; v), since pn ru n = 0 on .

Instead of essential boundary conditions (which suppose that the velocity profile is known), one can also decide to impose a pressure force (Neumann boundary condition) on the artificial borders which close the domain. Then we consider: ru n pn = pn on ; = fin; outg.

### Natural boundary conditions involving pressure drop, with energy conservation

We have detailed in Section 2.2.2 one way to solve the lack of energy conservation, with a constraint over the velocity profile, see (2.2.9). One can imagine that the difficulty in the existence and uniqueness studies can be overcome also by changing the variational formulation of the problem in order to obtain energy conservation, even with natural boundary conditions. The formulation we review in this section are based on an equivalent (in the continuous field) form of the advective term. Of course, changing the variational formulation also changes the problem that is being solved (since the associated boundary conditions change) and may makes it unsatisfactory from a modeling point of view.

Total pressure The nonlinearity in the Navier–Stokes equations can be written in several ways, which are equivalent in the continuum formulation for regular fields (since ru = 0). One leads to an energy-preserving formulations using the identity r( 1 2 juj2) = u (tru). Then we can write the momentum equation of Navier–Stokes system as (see [70]): @tu + (u r)u u t(ru) u = r(p + 2 juj2) := rptot.

#### Numerical treatment, numerical behaviour VS suitable modeling

We have seen that the nonlinearity in the Navier–Stokes equations can be written in several ways, which are equivalent in the continuum formulation (since r u = 0), but which lead to different discrete forms. Indeed, in a discrete framework, the free-divergence equation is only weakly enforced, then we do not have an exact discrete free-divergence velocity. Moreover, the divergence of the discrete velocity may grow large enough and cause significant differences between different schemes. We will describe three forms in a discretized framework: the basic one (@tu+(u r)u), the total derivatives (Du Dt ) and one which conserves the energy (@tu + (u r)u u t(ru)). We will use a common test case: we solve the Navier–Stokes equations in a bifurcation, with a natural Neumann boundary condition at the inlet, and with free outlet boundary conditions at the outlet. We use P2=P1 approximation, pin(t) = 10 sin(t), and we run each test case during 5 seconds. Computation have been performed with the software Felisce [39], following the approach that we are going to present now.

**Precisions on all the test cases shown in this section Used meshes: bifurcations**

In all the simulations, we use several bifurcation meshes, see Figure 2.3. The mother branch has a diameter equal to 8:103 m. We note hmax the mesh size. In the Table 2.1, we give the main characteristics of the meshes used in the simulations, with the numbers of degrees of freedom if one uses a P2=P1 approximation. The geometry can be seen as the beginning of the respiratory tract. Indeed, the airways can be considered as the dyadic tube network, see [126]. The blood arteries were also considered as a network, see [43].

**Finite element discretization of the convective term**

The nonlinearity in the Navier–Stokes equations can be written in several ways, which are equivalent in the continuum formulation of the Navier–Stokes equations (since ru = 0), but which lead to different discrete formulations with different algorithmic costs, conserved quantities, and approximation accuracy ([59, 64]).

**Table of contents :**

Remerciements

Résumé

Abstract

Table des matières

**1 Introduction générale **

1.1 Le système respiratoire : un peu de physiologie

1.2 Quelques modèles

1.3 Problématique mathématique et numérique

1.4 Problématique de modélisation

1.5 Problématique de validation des modèles et méthodes numériques

1.6 Résumé détaillé du manuscrit

**2 Artificial boundaries and formulations for the incompressible Navier– Stokes equations: applications to air and blood flows. **

2.1 Introduction

2.2 A theoretical overview

2.3 Numerical treatment, numerical behaviour VS suitable modeling

2.4 Conclusion

**3 Numerical stability study of a multi-dimensional modelling of 3D airflows and blood flows. **

3.1 Introduction

3.2 Modelling of airflows and blood flows

3.3 Numerical analysis: treatment and stability study

3.4 Scientific computing: numerical stability observations

3.5 Conclusion

3.6 Appendix: to sum up the obtained estimates Table des matières

**4 Modeling of the flow limitation phenomenon in the human respiratory tract during forced expiration**

4.1 Introduction

4.2 Viscous mechanisms

4.3 Bernoulli effects

4.4 Gathering of the two mechanisms

4.5 Conclusion

**5 Flow through a bend: comparison between numerical simulations and experiments **

5.1 Introduction

5.2 Methods

5.3 Results and discussion

5.4 Conclusion

**6 Conclusions générales et perspectives **

6.1 Conclusions

**6.2 Validation physiologique du modèle de ventilation considéré dans cette thèse : des simulations réalistes, même pour des cas pathologiques**

6.3 Perspectives

Annexes

**A Implémentation numérique **

A.1 Cadre de développement : la bibliothèque FELiScE

A.2 Schémas numériques

A.3 Résolution

A.4 Comparaison du code à la littérature

A.5 Comparaison du code à l’expérience

**B Unités, ordres de grandeur, conversion **

B.1 Unités

B.2 Ordres de grandeur

B.3 Conversions

12 Table des matières

**C Maillages utilisés **

C.1 Tube

C.2 Bifurcation

C.3 Bronches

C.4 Aorte

C.5 Coude

**Bibliographie **