Natural boundary conditions involving pressure drop, with-out 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.
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 (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  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 ) 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 = p n on ; = fin; outg: (2.2.5).
In all the chapter, we use a constant pressure p on all . This problem is called the pressure drop problem in . Variational form Considering (2.2.5), R (pn ru n) v can be replaced by the following forms on the right-hand side: ‘ : H1( ) ! R Z p v n.
Numerical treatment, numerical behaviour VS suit-able 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 (DDut ) 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 bifur-cation, 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 , 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 8:10 3 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 begin-ning of the respiratory tract. Indeed, the airways can be considered as the dyadic tube network, see . The blood ar-teries were also considered as a network, see .
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 r u = 0), but which lead to diﬀerent discrete formulations with diﬀerent algorithmic costs, conserved quantities, and approximation accuracy ([59, 64]).
Basic formulation of the convective form. (Formulation A, see Section 2.2.2) We saw there were two diﬀerent ways to write the variational problem (considering or not free-divergence test functions, see problems P2.2.3 and P2.2.4). Here, we focus on Problem P2.2.3. To approximate it, one has to consider the subspace Mh which is a finite element approximation of M. Then the approximated problem becomes: for each t 2 [0; T ], seek uh( ; t) 2 Vh and ph( ; t) 2 Mh such that: 8 dt (uh(t); vh) + b(uh(t); uh(t); vh) > d vh Vh; t (0; T ); +a(uh(t); vh ) + d(vh; ph(t)) = ‘in(vh) + ‘out(vh); > 8 2 2 > > > d( uh (t); q )=0; 8 q M ; t (0; T ); < h h 2 h 2.
Comparison between the diﬀerent stabilization methods
In Figure 2.20, we gather the flux at the entrance obtained throughout the chapter. In Figure 2.21, we do the same but with test-cases involving higher inlet pressures. If one uses the stabilization detailed in Section 2.3.4 with a too high parameter, there is a poor agreement between the obtained solution and the others. Stability is thus achieved at the expense of accuracy. This method can modify a lot the solution whereas the streamline diﬀusion method and the characteristics method lead to similar numerical solutions. Actually, we assume that the characteristics method introduces a diﬀusion along the characteristic lines, as using the streamline diﬀusion method.
Table of contents :
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.
2.2 A theoretical overview
2.3 Numerical treatment, numerical behaviour VS suitable modeling
3 Numerical stability study of a multi-dimensional modelling of 3D airflows and blood flows.
3.2 Modelling of airflows and blood flows
3.3 Numerical analysis: treatment and stability study
3.4 Scientific computing: numerical stability observations
3.6 Appendix: to sum up the obtained estimates
4 Modeling of the flow limitation phenomenon in the human respiratory tract during forced expiration
4.2 Viscous mechanisms
4.3 Bernoulli effects
4.4 Gathering of the two mechanisms
5 Flow through a bend: comparison between numerical simulations and experiments
5.3 Results and discussion
6 Conclusions générales et perspectives
6.2 Validation physiologique du modèle de ventilation considéré dans cette thèse : des simulations réalistes, même pour des cas pathologiques
A Implémentation numérique
A.1 Cadre de développement : la bibliothèque FELiScE
A.2 Schémas numériques
A.4 Comparaison du code à la littérature
A.5 Comparaison du code à l’expérience
B Unités, ordres de grandeur, conversion
B.2 Ordres de grandeur
12 Table des matières
C Maillages utilisés