b) Videojuegos

From a physical point of view, an innite velocity amplitude or an innitely small length scale is not meaningful : it breaks conservation principles or it is too abstract : a quantity can be very large or very small compared to another but not innitely large or small. If singularities cannot occur in real ows, is it physically relevant to study the Euler and Navier-Stokes singularities ? Is it really a physical problem or only mathematical one ? Would the existence of the Navier-Stokes and Euler singularities have an impact on a real turbulent ow ?

3.3.1 Navier-Stokes case

The existence of singularities in a mathematical solution to the 3D incompressible Navier-Stokes equations would lead to the invalidation of the assumptions used to derive those equations from the compressible equations. Therefore, the existence of a singularity in a real ow modelled by such a solution is not implied, even if there would probably be an impact on this real ow.

Indeed, in the case of a singularity occuring in the solution of the 3D INSE, the velocity norm would increase to innity and a real ow modelled by this solution would become compressible. Therefore, the emergence of a singularity in the real ow is not implied at all as the 3D INSE would stop being a proper model. The real ow would probably be impacted though but would display only prints or remainings of the singularity and not a genuine one. The singularities problem would be shifted to the compressible Navier-Stokes equations, in which the compressible eects would maybe

prevent the apparition of a singularity. Note that even if the compressible Navier-Stokes equations could develop singularities in nite time, they would also stop being a proper model for a real ow as the scale separation assumed to derive them would not be achieved anymore. This suggests that the existence of a singularity in the solution of a mathematical model should be seen as an indication that this model is not physically relevant anymore and that additional physics should be added. Other examples exist : for instance, in fracture mechanics, the elasticity equations and the special boundary conditions near a crack tip lead to a divergence of the stress at the crack tip, the energy remaining all the same bounded. However, in a real material the stress does not take innite values : when it reaches some threshold value, the material does not follow the linear elastic equations anymore ; damage develops and the material will break. Everything happens as if the singularity was replacing a physical phenomenon not accounted for by the model. In the case of the INSE, the possible singularities would even have their own dissipation term, accounting for the dissipation due to the missing phenomena : Duchon and Robert showed in [Duchon and Robert, 2000] that a new dissipation term appears in the energy budget of the weak solutions of the INSE, which can be non-zero in the case of a singularity.

As a conclusion, the existence of singularities in the solutions of the INSE is of physical importance for two reasons. First, it should have an impact on real ows which would display prints of singu-larities, and second, it would invalidate the INSE as a proper model for incompressible ows. As the singularities should be very scarce, the INSE would probably remain a proper model most of the time though.

3.3.2 Euler case

Contrary to the 3D INSE, the 3D incompressible Euler equations are not a proper model for real viscous ows, even without singularities, because they do not feature any viscous eect. However, the existence of singularities in the solutions of the 3D incompressible Euler equations may have an impact on the solutions of the 3D INSE, and thus on real ows.

Indeed, in the Navier-Stokes equations, the viscous eects are negligible at large scales. The large-scale behaviour of the 3D incompressible Euler and Navier-Stokes equations should therefore be similar, and, if singularities were developing in the Euler equations, they may also start forming in the INSE before evolving dierently. They would therefore have an impact on the INSE that could be measured. The following reasoning, inspired by the multifractal model [Parisi and Frisch, 1985], shows what this impact could be.

Let u be a solution of the incompressible Navier-Stokes equation 1.9 and assume that it is locally h-Hölder continuous, i.e. behaving as follows :

||δu|| = ||u(x + `) − u(x)|| ∝ ||`||^h (1.18) We can then reproduce the energy cascade reasoning with δu ∝ `<sup>h</sup> instead of δu ∝ `^1/3. The Hölder condition leads to the following estimates of the viscous dissipation and inertial energy terms :

ν∂_iu_j∂_iu_j ≈ νδu²

`<sup>2</sup> ≈ ν`^2h

`<sup>2</sup> ≈ ν`^2h−2 (1.19)

uj∂j

u²_i 2 ≈ `^3h

` ≈ `^3h−1 (1.20)

Both terms are comparable when `<sup>3h−1</sup>≈ ν`^2h−2 i.e. when ` ≈ ν<sup>1/(1+h)</sup>. We dene ηh = ν<sup>1/(1+h)</sup>. All this can be interpreted as a local cascade with a particular scaling : there is an inertial range with δu ∝ `^h in which viscous eects are negligible and where energy is transferred to smaller scales.

3. THE POSSIBLE SINGULARITIES 31 The dissipative scales are of the order of ηh. The viscous dissipation absorbs the inter-scale transfer and no structures smaller than ≈ ηh can develop ; therefore, the velocity eld is smooth below this scale : δu(`) ∝ `. For h = 1/3, we nd the usual scaling, in particular, η1/3 is the Kolmogorov scale.

However, for other h, the situation is dierent ; for h < 1/3, ηh is smaller than the Kolmogorov scale.

According to this reasoning, the event {∃A ⊂ Rⁿ,∃x ∈ A| ∀y ∈ A, ||v(x) − v(y)|| ∝ ||x − y||^h} with −1 < h < 1 cannot occur in a solution of the 3D INSE : indeed, the viscosity would smooth the velocity eld below ηh and the Hölder condition ||v(x) − v(y)|| ∝ ||x − y||^h would hold only for ||x − y|| > η^h. However, it could occur in a solution to the 3D incompressible Euler equation where there are no viscous eects. We therefore call it a Euler-type singularity, as opposed to a Navier-Stokes-type singularity, if h = −1 : even viscous eects cannot smooth the velocity eld as the regularization scale is η−1 = 0, and the velocity eld is therefore singular. The singular elds shown in gure 1.4 are Navier-Stokes-type singularities. Euler-type singularities (−1 < h < 1) could be an example of a singularity of the 3D incompressible Euler equations which have an impact on the 3D INSE : such events developing in the 3D INSE would lead to a multiplicity of dissipative scales, in particular smaller than the Kolmogorov scale. This would be quite a change in the classical picture of turbulence. The existence of too small length scales could however question the scale separation required for the derivation of the Navier-Stokes equations. Such events would also lead to a multiplicity of scalings, and would impact the statistics of the velocity eld. This is the idea behind the multifractal model, which provides good t of experimental data. Also, according to this model, the smaller h, the less probable the occurence of a locally h-Hölder continuous velocity

eld. This suggests that Navier-Stokes-type singularities are very scarce, in accordance with the result of [Cafarelli et al., 1982], and rules out too small ηh, ensuring the scale separation. But it is not known yet whether the previous reasoning corresponds to the reality or not. The multifractal model also has a probabilistic formulation which does not require such a phenomenology. Besides, the organization of the ow as a juxtaposition of places where δu ∝ `^h with dierent h was never directly evidenced. One should therefore be careful with this representation. In particular, the above reasoning does not constitute a rigorous proof that singularities with h > −1 cannot occur in a solution of the Navier-Stokes equations.

Summary

In this chapter we introduced the 3D incompressible Navier-Stokes equations, the partial dierential equations which govern the motion of incompressible uids and which therefore feature turbulence.

The regularity of these equations, as well as the 3D incompressible Euler equations, are still open mathematical problems. The possible singularities are incidentally related to intermittency and to the dissipation anomaly, two turbulence features which are not clearly connected with the 3D INSE yet.Even if singularities have no physical meaning and are not to occur in the physical world, their mathematical existence should have an impact on real ows. Therefore, we claim that studying a real ow allows to give insight into the possible singularities of the Euler and Navier-Stokes equations, or at least their formation. This is detailed in the next chapter.

Chapter 2

The problem of singularities addressed by the experiment

In this chapter we explain how we tackle the problem of singularities with an experimental approach.

We use a detection criterion which is based on the work of Duchon and Robert [Duchon and Robert, 2000] in order to detect prints of singularities, i.e. places that might correspond to the formation of singularities or to aborted singularities. This criterion was already introduced in [Kuzzay et al., 2017] and applied to 2D-3C data in [Saw et al., 2016] ; in this thesis we will apply it to 3D-3C data. The detection principle can be applied to any velocity eld, obtained either by a numerical simulation or an experimental measurement ; we then compare both approaches and explain why we chose an experimental approach. Finally, we detail the outline of the method used in this thesis.

1 Prints of singularities as extreme events of inter-scale transfer rate at small scale

Our detection method is based on the work by Duchon and Robert [Duchon and Robert, 2000]. A singularity corresponds to an innite renement of scales, and this mathematical work states for-mally that such a singularity may come along with a non-zero inter-scale transfer down to innitely small scales. This inter-scale transfer interpretation establishes a link with the LES equations.