domain: info math

text:
Jean-Daniel Boissonnat
email: jean-daniel.boissonnat@inria.fr

Mathijs Wintraecken
email: mathijs.wintraecken@inria.fr

A T E X

Keywords: Reach, Manifolds, surfaces, comparison theory MSC Classification: 53-08, 53A99, 68U05, 65D18

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Introduction

Motivation. The reach of subsets of Euclidean space was introduced by Federer [START_REF] Federer | Curvature measures[END_REF] and can be defined from the medial axis introduced later by Blum [START_REF] Blum | A Transformation for Extracting New Descriptors of Shape[END_REF] The reach of subsets of manifolds (although it has also been studied by Federer). The medial axis of a set S in Euclidean space consists of the points for which there is no unique closest point on S. The reach of S, denoted rch(S), is defined as the distance between the medial axis and the original set. It is therefore the largest distance such that if a point p ∈ R d is moved at most this distance from S, then there exists a unique closest point on π S (p) ∈ S. Here we write π S for the closest point projection onto S.

The reach is used to quantify the complexity of geometric sets of R d . In particular, from the reach, one can infer bounds on the local curvature [START_REF] Boissonnat | The reach, metric distortion, geodesic convexity and the variation of tangent spaces[END_REF] of submanifolds. The reach also provides more global information, such as how close different parts of a set S lie to one another. It has been extensively used to define sampling conditions on submanifolds of R d , to quantify how difficult it is to infer their topology (see for example [START_REF] Boissonnat | Geometric and Topological Inference[END_REF][START_REF] Wang | Topological inference of manifolds with boundary[END_REF][START_REF] Brown | Sheaf-Theoretic Stratification Learning[END_REF][START_REF] Bendich | Stratification learning through homology inference[END_REF][START_REF] Bendich | Inferring local homology from sampled stratified spaces[END_REF]), to triangulate them (see for example [START_REF] Boissonnat | Geometric and Topological Inference[END_REF][START_REF] Boissonnat | Meshing of surfaces[END_REF][START_REF] Cheng | Delaunay Mesh Generation[END_REF][START_REF] Dey | Curve and Surface Reconstruction: Algorithms with Mathematical Analysis[END_REF][14]), or learn them (see for example [START_REF] Fefferman | Fitting a manifold of large reach to noisy data[END_REF][START_REF] Aamari | Statistical query complexity of manifold estimation[END_REF][START_REF] Aamari | Stability and minimax optimality of tangential Delaunay complexes for manifold reconstruction[END_REF][START_REF] Fefferman | Reconstruction and interpolation of manifolds. I: The geometric Whitney problem[END_REF][START_REF] Sober | Manifold approximation by moving least-squares projection (MMLS)[END_REF]).

Due to its importance and ubiquity, the reach has been redefined several times and appears under various names in the literature. It was called the condition number by Niyogi, Smale and Weinberger [START_REF] Niyogi | Finding the homology of submanifolds with high confidence from random samples[END_REF]. A local version of the reach, named the local feature size, was introduced in the computational geometry community by Amenta and Bern in their seminal work on surface reconstruction [START_REF] Amenta | Surface reconstruction by Voronoi filtering[END_REF]. It is the distance of a point to the medial axis and was denoted reach(S, p) by Federer [START_REF] Federer | Curvature measures[END_REF]Definition 4.1].

Extending the notion of reach to subsets S of Riemannian manifolds M is a question that immediately shows up when one considers triangulating manifolds with boundary, submanifolds of Riemannian manifolds, or stratified manifolds, as required in dynamical systems, physics, and chemistry. Examples of stratified manifolds are the invariant sets that appear in dynamical systems [START_REF] Henderson | Multiple parameter continuation: computing implicitly defined k-manifolds[END_REF] and the conformation spaces of molecules [START_REF] Martin | Topology of cyclo-octane energy landscape[END_REF].

One particular motivation is the search for anisotropic triangulations for numerical partial differential equations, see e.g. [START_REF] Babuvška | Error estimates for adaptive finite element computations[END_REF][START_REF] Wang | A discontinuous galerkin method for the navierstokes equations on deforming domains using unstructured moving spacetime meshes[END_REF][START_REF] Abedi | Spacetime meshing with adaptive refinement and coarsening[END_REF][START_REF] Von Danwitz | Simplex spacetime meshes in compressible flow simulations[END_REF][START_REF] Labelle | Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF][START_REF] Apel | Anisotropic interpolation with applications to the finite element method[END_REF][START_REF] Habashi | Anisotropic mesh adaptation: towards userindependent, mesh-independent and solver-independent CFD. Part I: general principles[END_REF][START_REF] Aguilar | Anisotropic mesh refinement for finite element methods based on error reduction[END_REF], based on Riemannian metrics as in [START_REF] Boissonnat | Anisotropic triangulations via discrete Riemannian Voronoi diagrams[END_REF] but with interfaces or boundaries. To triangulate these complicated spaces one needs to understand the geometry of the interfaces and boundaries with respect to the Riemannian metric.

Another motivation comes from the reconstruction of submanifolds with boundary embedded in high dimensional Euclidean space. Working with such data one generally faces the curse of dimensionality, which could be avoided by ignoring the embedding completely and only working with the intrinsic geometry.

Contributions. In the early 1980s Kleinjohann [START_REF] Kleinjohann | Convexity and the unique footpoint property in Riemannian geometry[END_REF] and Bangert [START_REF] Bangert | Sets with positive reach[END_REF] extended the definition of the reach to subsets S of Riemannian manifolds M, see also [START_REF] Kleinjohann | Nächste Punkte in der Riemannschen Geometrie[END_REF]. The reach of a submanifold of a Riemannian manifold is similar to the definition in Euclidean space, which is roughly the distance to the medial axis, i.e. the set of points of M that have more than one closest point on S. Kleinjohann was interested in (locally) convex subsets of Riemannian manifolds and their differential geometric properties, and Bangert studied the relation between the convex functions and (locally) convex sets. Their definition of the reach is not fully satisfactory for the applications mentionned above. Indeed, consider the cylinder S 1 × R = M and the submanifold S 1 × {0} = S. According to the definition of Kleinjohann and Bangert, the reach of S 1 × {0} is infinite. However, at scales larger than 2π the 'balls' tangent to S 1 ×{0} are no longer topological balls, which is very inconvenient for practical applications.

We will introduce a slight variant of the reach for submanifolds of arbitrary smooth Riemannian manifolds. We add the condition that we look at neighbourhoods smaller than the (global) injectivity radius. 1 This assumption makes comparison results from both Riemannian geometry and computational geometry more straightforward to apply.

Our main result, Theorem 4.11, allows one to lift via the exponential map the results known on the reach of subsets of R d to submanifolds of Riemannian manifolds. More specifically, given p ∈ M and q ∈ S, we give explicit bounds on the local feature size (local reach) of its lifting to the tangent space via the inverse exponential map (exp -1 p (S)) at q, assuming that rch M (S) and the geodesic distance d M (p, q) are bounded. We stress that the explicit bounds are of key importance, the fact that the local feature size is non-zero is easy to see, i.e. it is a direct consequence of smoothness. This quantified approach to the problem also distinguishes us from previous results in the literature [START_REF] Kleinjohann | Convexity and the unique footpoint property in Riemannian geometry[END_REF][START_REF] Bangert | Sets with positive reach[END_REF][START_REF] Kleinjohann | Nächste Punkte in der Riemannschen Geometrie[END_REF] where the focus was on existence. Even though the expression for our bound remains non-trivial we provide closed expressions for the bounds. The computation of our bounds rests on two pillars: Comparison theory in particular the Rauch comparison theorem and the Kaul's bounds (see Section 4.2) and Federer's result on the behavior of the reach under an ambient diffeomorphism (Theorem 4.9). We note that our bounds could be improved (by using Kaul's bounds directly instead of a simplified version, see Section 4.2).

We see our bounds as a first major step in the long term quest of efficient algorithms to triangulate stratified Riemanniann manifolds. This will allow to extend the work in [14] by locally lifting a submanifold of a Riemannian manifold M to the tangent space of M using the exponential map, see Future work in Section 5. A similar but more complicated approach can be used for manifolds with boundary and stratifolds. The reach of S and exp -1 (S) will be used to quantify the conditions under which we obtain a triangulation.

The reach

Similar to the definition for manifolds embedded in Euclidean space we define the reach for compact subsets S ⊂ M, that is we make the global assumption that S is compact.

Definition 2.1 (The reach) We let the medial axis ax M (S) be the set of points in M that do not have a unique closest point on S, with respect to the Riemannian metric. We denote the projection of a point x in M on the closest point on S by π S (x). If 1 Recall that the injectivity radius is the largest radius such that the exponential map restricted to a ball of this radius is still a homeomorphism onto its image. The exponential map exp p : TpM → M sends vectors to geodesics through p while preserving lengths and angles at p. The reach of subsets of manifolds x lies in the medial axis this map is set valued. The pre-reach prch M (S) is then the shortest distance between ax M (S) and S. We now define the reach rch M (S) to be

min{prch M (S), ι M }, (1)
where ι M is the injectivity radius of M.

The injectivity radius is the largest radius r such that for any x, exp x restricted to the open ball centred at x with radius r in T x M is a diffeomorphism onto its image. Adding this bound to our definition of the reach is essential, because we would like to have that a tangent ball to S is indeed a topological ball.

We can also define a local version of the reach we call local feature size in accordance with the terminology used for subsets of Euclidean space [START_REF] Amenta | Surface reconstruction by Voronoi filtering[END_REF]. For a point y ∈ S, it is defined as the minimum of the injectivity radius at y and of the distance of y to ax M (S).

Remark 2.2

In the mathematics literature, the closure of the medial axis is sometimes called the cut locus in the context of Riemannian manifolds, see [START_REF] Chazal | Stability and finiteness properties of medial axis and skeleton[END_REF]Section 2] or [START_REF] Lieutier | Any open bounded subset of R n has the same homotopy type as it medial axis[END_REF] (often the cut locus only refers to the medial axis of a single point in a Riemannian manifold). We have chosen to use the terminology and concept most common in the computational geometry community in view of future applications. Bangert [START_REF] Bangert | Sets with positive reach[END_REF] referred to what is here called the pre-reach as the reach. Kleinjohann seems to have been unaware of the work of Federer when writing [START_REF] Kleinjohann | Convexity and the unique footpoint property in Riemannian geometry[END_REF].

Throughout this paper we'll assume that M is smooth, by which we mean at least C 3 . This is essential because we rely on the result of Kaul [START_REF] Kaul | Schranken für die Christoffelsymbole[END_REF], see Section 4.2 below, which is based on bounds on the derivative of the Riemann curvature tensor. Because the Riemann curvature tensor involves second order derivatives the manifold has to be C 3 . However, we conjecture that most of the results go through (with some minor adjustments) for C 2,1 manifolds, that is manifolds whose second order derivative is Lipschitz.

Smooth submanifolds have positive reach

In this section we restrict ourselves to a smooth submanifold S of a Riemannian manifold M. It suffices for the submanifold S to be C 2 , in this section. In Corollary 4.13 we'll see (by reduction to the Euclidean case) that this can be weakened to C 1,1 , however this is far from trivial. We shall denote the normal space of S at a point x by N x S, and the bundle by N S.

We now first need a counterpart of Theorem 4.8.8 of Federer [START_REF] Federer | Curvature measures[END_REF] in the setting of smooth submanifolds S of Riemannian manifolds: Theorem 3.1 (Tubular neighbourhood) Let B NpS (r), be the ball of radius r centred at p in the normal space NpS ⊂ TpM of a C 2 manifold with reach rch M (S), where r < rch M (S). For every point x ∈ exp p (B NpS (r)), we have π S (x) = p.

Variants of this result are well known in differential topology and Riemannian geometry, but a proof is included for completeness. Kleinjohann [START_REF] Kleinjohann | Nächste Punkte in der Riemannschen Geometrie[END_REF]Section 3] proves related statements, but uses a somewhat different definition of normal. The proof of Theorem 3.1 is not so difficult and mainly follows Hirsch [START_REF] Hirsch | Differential Topology[END_REF]Section 4.5] and Spivak [38, With this lemma we can prove an embedding result, for which we have to make the following definition: Definition 3.3 Let N Bϵ denote the ϵ-neighbourhood in N S of S, that is the neighbourhood of all points closer than ϵ to S, where we identify S ⊂ N S via the zero section. Moreover write f : N Bϵ → M, for the map defined by sending S ⊂ N S to S and fiberwise sending NpS to exp p (NpS).

Lemma 3.4

There exists an ϵ > 0 such that f : N Bϵ → M is a (global) homeomorphism onto its image, that is an embedding.

The proof combines arguments from Section 4.5 of [START_REF] Hirsch | Differential Topology[END_REF] with Appendix I of Chapter 9 of [START_REF] Spivak | A Comprehensive Introduction to Differential Geometry: Volume I[END_REF], where small variations of this statement can be found.

Proof of Lemma 3.4 We first note that the fiberwise restriction of f to B NpS (r) is a diffeomorphism for each r < ι M . Because M is smooth, f is smooth and we can consider the derivative T (p,0) f at a point (p, 0) ∈ N S. The tangent space splits as follows T (p,0) N S = TpS ⊕ NpS. T (p,0) f is the identity if restricted to the tangent space of M as well as to a fiber. This gives that f is an immersion and in particular a local homeomorphism on its image because the codimension is zero. Due to Lemma 3.2, f : N Bϵ → M is injective for some sufficiently small ϵ > 0. □ Note that in the proof of Lemma 3.4, we used the fact that S is C 2 because we considered the derivative of the normal (note that the tangent space is found by taking first derivatives and therefore the derivative of the normal involves second order differentiation).

We now define N S(r) fiberwise as those points in N p S that are at a distance r from p. We refer to r as the radius of the tubular neighbourhood. For any smooth manifold S embedded in M and x ∈ M, we know that the minimizing geodesic from x to π S (x) is normal to S at π S (x), as a direct consequence of the Gauss lemma (the Gauss lemma that refers to the exponential map). It follows that, for all ϵ such that f : N B ϵ → M is a homeomorphism, each point in f (N S(r)) is a distance r from S, for all 0 < r < ϵ.

Because S is compact, so is N S(r) as is the closed r-neighbourhood N B r . Clearly f is continuous. In general, a continuous bijection from a compact to a Hausdorff space is a homeomorphism. This means that the map f from both N S(r) and the closed r-neighbourhood N B r to their images are homeomorphisms, that is embeddings, if and only if f from both N S(r) and N B r are injective.

We define the reach rch M (S) of the submanifold S ⊂ M to be the infimum of all radii r such that f from both N S(r) and N B r are no longer injective. We have that rch M (S) ≥ ϵ > 0. By the same argument as above, the distance from a point in N S(r) to S is r for all r such that f restricted to N S(r) is injective. Or equivalently, the distance from a point in f (N S(r)) to S is r, for all r < rch M (S). Because the distance function to any closed set is continuous (it is even Lipschitz, see [3, Theorem 4.8(1)]), this in fact holds also for r = rch M (S).

In summary we have, Lemma 3. [START_REF] Boissonnat | The reach, metric distortion, geodesic convexity and the variation of tangent spaces[END_REF] The reach rch M (S) is equal to the infimum over all radii r, with r ≤ ι M , such that f restricted to N S(r) and N Br are no longer homeomorphisms to their images. For all r ≤ rch M (S), all points in f (N S(r)) are a distance r from S.

In particular we have proven Theorem 3.1.

Here and in the following section we'll often speaking about (open or closed) ball B being tangent to a submanifold S ⊂ M. By this we mean that ∂B and S have a point p in common such that the tangent spaces T p S ⊂ T p M and T p ∂B ⊂ T p M satisfy T p S ⊂ T p ∂B. Note that Theorem 3.1 immediately yields, Corollary 3. [START_REF] Boissonnat | Geometric and Topological Inference[END_REF] The interior of any ball B ⊂ M that is tangent to S at p∈ S and whose radius r satisfies r ≤ rch M (S) does not intersect S.

Proof Let r < rch M (S). Suppose that the intersection of S and the interior of the ball is not empty, then π S (c) ̸ = p, contradicting Theorem 3.1. The result for r = rch M (S) now follows by taking the limit. □

Main result: Bounds on the local feature size

Our main result is to link our definition of the reach with the local feature size of submanifolds of Euclidean space in an explicit and quantified manner.

Namely given a set of positive reach S in a Riemannian manifold M, p ∈ M and q ∈ S, we bound the (Euclidean) local feature size of exp -1 p (S) ⊂ T p M at q, assuming that the geodesic distance d M (p, q), the curvature and derivatives of the curvature are bounded.

As a nice byproduct of the main result of this paper and [START_REF] Lieutier | Embedded topological manifolds of positive reach with boundary of positive reach are C[END_REF], see also [START_REF] Lytchak | On the geometry of subsets of positive reach[END_REF][START_REF] Lytchak | Almost convex subsets[END_REF], we see that a topologically embedded submanifold (with boundary) of a Riemannian manifold has positive reach (and a positive reach boundary) if and only if it is C 1,1 (with a C 1,1 boundary). Lytchak's results [START_REF] Lytchak | On the geometry of subsets of positive reach[END_REF][START_REF] Lytchak | Almost convex subsets[END_REF] do include the Riemannian case, but the work by Lytchak [START_REF] Lytchak | On the geometry of subsets of positive reach[END_REF][START_REF] Lytchak | Almost convex subsets[END_REF] relies in an essential manner on results from CAT(k)-theory, while the reduction in this paper requires only some minimal bounds on Christoffel symbols and is quantified.

Approach

The lower bounds on the reach of exp -1 p (S), in terms of the reach of S and geometric properties of the manifold, follow by considering tangent balls to S that do not contain points of S, from various viewpoints, namely from the manifold and from various tangent spaces. The argument will be explained in detail using Figure 1.

M S c q TcM TpM exp -1 p exp -1 c exp -1 p • exp c c exp -1 c (q) exp -1 c (S) p exp -1 c (p) exp -1 p (S) exp -1 p • exp c (Bc) exp -1 p (q) B Bc = exp -1 c (B)
Fig. 1 An overview of the approach.

Let B be a geodesic ball in M centered at c, tangent to S at a point q and with radius the reach of S. As in Euclidean space and by Corollary 3.6, B has an empty intersection with S. B is indicated in red in the top of Figure 1.

The exponential map exp c gives coordinates on a neighbourhood of the manifold, as does exp p . We will use the inverse of the exponential maps exp c The reach of subsets of manifolds and exp p , and write B c = exp -1 c B and B p = exp -1 p B for the images (in T c M and in T p M respectively) of the geodesic ball B. Note that B c is an Euclidean ball which implies that its reach is equal to its radius, which is also the (geodesic) radius of B. See bottom left part of Figure 1. Note also that B c is tangent to S c = exp -1 c (S) and that B p is tangent to S p = exp -1 p (S). The composition exp -1 p • exp c gives a transformation between the two coordinate neighbourhoods, as indicated by the arrow from the bottom left to right in Figure 1.

We stress that we follow this convention for the notation of balls: geodesic balls on the manifold are denoted by calligraphic capital letter B without subscript. The image of a set under the inverse exponential map to a tangent space are denoted in the same way as the original set, but with an extra subscript indicating the point where the tangent space is taken. Euclidean balls in the tangent space will be denoted by lower case bold letters b p , with a subscript indicating the point where we take the tangent space. These balls are (except in very particular cases) not the image of a geodesic ball. The one exception to this rule is the ball B c , which is both the image of a geodesic ball and a Euclidean ball and therefore is denoted by a capital letter, like the goedesic balls, but bold like the Euclidean balls. All balls can be assumed to be open.

Thanks to the Toponogov or Rauch comparison theorem and a higher order variant (namely Kaul's bounds on the Christoffel symbols, see Section 4.2 below), we have bounds on the metric as well as on the derivatives of the metric, both expressed in the coordinates induced by the exponential maps exp p and exp c . The bounds on the metric and its derivatives can then be used to give bounds on the first and second order derivatives of the transformation exp -1 p • exp c . Thanks to a result by Federer [START_REF] Federer | Curvature measures[END_REF] one can find a bound on the reach after the transformation, based on the reach of the original. This gives bounds on the reach (in T p M) of B p , that is on the reach of exp -1 p • exp c (B c ). Consider the Euclidean ball b p ⊆ B p ⊂ T p M with radius rch(B p ) that is tangent to ∂B p at exp -1 p (q), where ∂ indicates the boundary. 2 The Euclidean ball b p is indicated in the bottom right of Figure 1, by the green Euclidean ball inside the red deformed ball. The Euclidean ball b p is also tangent to S p at exp -1 p (q) and, since the interior of B does not intersect S, B p does not intersect S p either nor does b p .

The reach of S p is lower bounded by the minimum radius of any such Euclidean ball b p , that is rch(S p ) ≥ min where the minimum over B is a minimum over geodesic balls in M of radius rch M (S) that are tangent to S at q. For the local feature size one takes q fixed instead of minimizing over S.

Outline. This section is subdivided as follows:

• In Section 4.2 we focus on Riemann normal coordinate systems. Thanks to standard comparison theorems such as the Toponogov or Rauch comparison theorem [START_REF] Heintze | A general comparison theorem with applications to volume estimates for submanifolds[END_REF][START_REF] Karcher | Riemannian comparison constructions[END_REF], we are able to give bounds on the metric in Riemann normal coordinates. The Riemann normal coordinates of a neighbourhood of p are those coordinates that are found by lifting the metric to the tangent space at p via the exponential map. The work by Kaul [START_REF] Kaul | Schranken für die Christoffelsymbole[END_REF] provides us bounds on the Christoffel symbols in the Riemann normal coordinate neighbourhood, and thus indirectly bounds on the derivative of the metric. • In Section 4.3 we'll study the coordinates transformation exp -1 c • exp -1 p , the bottom arrow in Figure 1. In Section 4.3.1, we first see how we can go from bounds on the metric in the Riemann normal coordinates to bounds on the coordinate transformation exp -1 c • exp -1 p . In Section 4.3.2, we'll be applying a result by Federer [START_REF] Federer | Curvature measures[END_REF] that will yield the reach.

Bounds on the metric and Kaul's bound on the Christoffel symbols

In this section we review the bounds on the metric and Kaul's bounds on the Christoffel symbols, see [START_REF] Kaul | Schranken für die Christoffelsymbole[END_REF]. The expressions for these bounds have been simplified by Von Deylen [44, Section 6], at the cost of weakening the bounds. We shall make use of his simplification.

As is standard in Riemannian geometry, the (coordinates of) vectors (elements of the tangent space or bundle) are indicated by v i (so that the vector v with coordinates v i is given by v = i v i ∂ i ) and the (coordinates of) covectors (elements of the cotangent space or bundle) by ω i (so that ω = i ω i dx i ). Similarly we follow the standard convention for all tensors, e.g. the (covariant) metric is denoted by g ij (even though in some sense it would be better to only refer to g = ij g ij dx i ⊗ dx j as the metric). Writing g = ij g ij dx i ⊗ dx j is compatible with the fact that the metric at a point is a bilinear form on the tangent space of that point. The inverse metric tensor or contravariant metric tensor is denoted by g ij , so that j g ij g jk = δ k j . If we want to emphasize the point where to consider the (inverse) metric we write g ij (x) and g ij (x) respectively. We employ similar notation for all other tensors. The standard Euclidean metric is given by δ ij . Informally one can think of tensors as generalizations of matrices (multi-linear maps), however in General Relativity and most of Riemannian geometry (covariant or contravariant) tensors also behave in a very specific way under coordinate transformations (using the Jacobian of the coordinate transformation and/or its inverse). We will encounter a number of multi-linear maps below, such as the Christoffel symbols, which are not tensors.

The norm with respect to the Riemannian metric is denoted by | • | g while the norm with respect to the Euclidean metric is denoted by | • | E . Distances on M will be denoted by d M . As before, ι M is the injectivity radius. The reach of subsets of manifolds The Christoffel symbols are

Γ κ µν (x) = λ 1 2 g κλ (x)(∂ µ g λν (x) + ∂ ν g λµ (x) -∂ λ g µν (x))
where ∂ µ denotes the partial derivative with respect to the coordinate x µ . As mentioned, Christoffel symbols are themselves not tensors. The Christoffel symbols are used to express the covariant derivative of a vector field v κ in local coordinates,

∇ ν v κ = ∂ ν v κ + µ Γ κ µν (x)v µ .
The Riemann curvature tensor in local coordinates will be denoted by R σ µνλ (x). We refer to Do Carmo [START_REF] Do Carmo | Riemannian Geometry[END_REF] and Spivak [START_REF] Spivak | A Comprehensive Introduction to Differential Geometry: Volume I[END_REF], as some of the standard texts introducing these concepts.

We now consider the Riemann normal coordinates around p:

x : (x 1 , . . . , x d ) → exp p (x i E i )
for some orthonormal basis E i of T M .

We shall now assume that the curvature and its derivative are bounded, that is in any normal coordinate system,

|R σ µνλ (x)| ≤ R max (2)
|∇ κ R σ µνλ (x)| ≤ R ∇ max . ( 3
)
We will use the following version of the Toponogov or Rauch comparison theorem.

Lemma 4.1 (Weak Rauch theorem, Lemma 6.8 of [START_REF] Von Deylen | Numerische approximation in Riemannschen mannigfaltigkeiten mithilfe des Karcher'schen schwerpunktes[END_REF])

If d M (p, x) ≤ r, with Rmaxr 2 ≤ π 2 4 and r ≤ ι M 2 , then |g ij (x) -δ ij | ≤ Rmaxr 2 in Riemann normal coordinates around p.
Usually the Rauch comparison theorem is stated in terms of the sectional curvature, however for our setting reformulating the statement in terms of bounds on the Riemann curvature is more convenient. Moreover the bounds of Kaul [START_REF] Kaul | Schranken für die Christoffelsymbole[END_REF] can be simplified (at the cost of weakening) to: Lemma 4.2 (Weak Kaul lemma, Lemma 6.9 of [START_REF] Von Deylen | Numerische approximation in Riemannschen mannigfaltigkeiten mithilfe des Karcher'schen schwerpunktes[END_REF])

If d M (p, x) ≤ r, with Rmaxr 2 ≤ π 2 4 and r ≤ ι M 2 , then κ,λ,µ,ν g κλ (x)Γ κ µν (x)v µ w ν u λ ≤ 10Rmaxr + 5R ∇ max r 2 ,
for all v, u, w ∈ TxM such that |u|g = |v|g = |w|g = 1 and using Riemann normal coordinates around p.

We now note that thanks to Lemma 4.1, we have that

||v| 2 g -|v| 2 E | = i,j g ij (x)v i v j -δ ij v i v j ≤ ij |g ij (x) -δ ij ||v i v j | ≤ ij R max r 2 |v i ||v j | = R max r 2 i 1|v i | 2 ≤ R max r 2 i 1 i |v i | 2 , = dR max r 2 |v| 2 E ,
where we used the Cauchy-Schwartz inequality. Combining this with Lemma 4.2, we see that

κ g κλ (x)Γ κ µν (x) ≤ 10R max r + 5R ∇ max r 2 (1 -R max d r 2 ) 3/2 . Using that ∂ ν g κµ (x) = λg κλ (x)Γ λ µν + g µλ (x)Γ λ κν
and taking absolute values, we find the following corollary

Corollary 4.3 If d M (p, x) ≤ r, with Rmaxr 2 ≤ π 2 4 and r ≤ ι M 2 , then |∂ν gκµ(x)| ≤ 20Rmaxr + 10R ∇ max r 2 (1 -Rmaxd r 2 ) 3/2 ,
using Riemann normal coordinates around p.

We now recall two results from linear algebra:

• Let E be a d × d-matrix, then (see for example (2.3.8) of [START_REF] Golub | Matrix Computations, 3rd edn[END_REF])

max ij |E ij | ≤ ∥E∥ 2 ≤ d max ij |E ij | (4)
• If G = I + E, where G and E are d × d-matrices, I denotes the identity matrix and ∥E∥ 2 ≤ 1, then (see for example [47, Section 5.8])

∥I -G -1 ∥ 2 ≤ ∥E∥ 2 1 -∥E∥ 2 , If now, |E ij | ≤ c, with dc < 1, then max i,j |(I -G -1 ) ij | ≤ dc 1 -dc .
With this result and Lemma 4.1, in particular setting g ij (x) = G and recognizing that We'll also make use of the following result [START_REF] Horn | Matrix Analysis[END_REF]Corollary 6.3.4], which we'll formulate as a lemma,

|E ij | = |g ij (x) -δ ij | ≤ R max r 2 ,

From bounds on the metrics to bounds on the coordinate transformations

The starting point of this section is the following: We are given a metric in Riemann normal coordinates at two different points. We want to study the coordinate transformation between these coordinates systems, based on our knowledge of the metric in these two coordinate systems.

In fact, we assume we have bounds on the first and second order derivatives of the metric in both coordinate systems. These bounds yield bounds on the first and second derivatives of the coordinates transformation. This can be understood by considering the limit case: Suppose both metrics are the Euclidean metric, then the transformation from one coordinate system to the other is a combination of a rotation and translation.

From bounds on the coordinate transformation, a result of Federer [START_REF] Federer | Curvature measures[END_REF] gives bounds on the reach of exp -1 p (B) where B is a geodesic ball centred at c with radius r, assuming that p and c are not too far from each other. Here we emphasize (as we noted before) that the radius of the ball B equals the radius of exp -1 c (B). We will consider a coordinate transformation from a coordinate system x to a coordinate system y. Because the emphasis is on coordinate transformations we'll follow a different convention in this section, and only this section, and use Latin indices. We'll use the indices a, b, c, e, f for y-coordinate system and the indices i, j, k, l, m for x and write

y a = i T a i x i + i,j Q a ij x i x j + O(x 3 ).
Here we assumed that the coordinate systems are chosen such that the origins are mapped to one another, which can be done without loss of generality. Q a ij is symmetric in i and j.

Here and in Subsection 4.3.1 all tensors (g ij , its inverse, its derivatives, etc.) will be evaluated at the origin (0) of the coordinate system unless specifically stated otherwise.

We have

g ij (x) = a,b g ab (y(x)) ∂y a ∂x i ∂y b ∂x j g ij (x) = g ij + k (∂ k g ij )x k + O(x 2 ) g ab (y) = g ab + c (∂ c g ab )y c + O(y) 2 y c = m T c m x m + O(x 2 ) ∂y a ∂x i = k T a i + Q a ik x k + O(x 2 ).
Combining these gives

g ij = i,j g ab T a i T b j , ( 5
)
∂ k g ij = a,b,c ∂ c g ab T c k T a i T b j + a,b g ab (Q a ik T b j + T a i Q b jk ).

Bounds on the transformations

With the concepts and notations developed in the previous section, we can give bounds on the transformation in terms of bounds on the metric and its derivatives. We find the following.

Lemma 4.7 Let x and y be two (smooth) coordinates systems for the same point on the manifold. By the smoothness assumption the metric and its derivatives in these coordinates systems are as follows

g ij (x) = g ij + k (∂ k g ij )x k + O(x 2 ) g ab (y) = g ab + c (∂cg ab )y c + O(y 2 ).
We assume further that:

• Any eigenvalue λ of g ij is bounded by |λ -1| ≤ A. • Any eigenvalue λ of g ab is bounded by | λ -1| ≤ B.
• The entries of g ef are bounded from above by 1 + C.

• For all i, j, k we also have that

|∂ k g ij | ≤ ∂g max,x
, and for all a, b, c, that |∂ a g bc | ≤ ∂g max,y .

Now we have that the coordinate transformation between x and y,

y a = i T a i x i + i,j Q a ij x i x j + O(x 3 ),
satisfies the following constraints:

• The Lipschitz constants, or the metric distortion of the linear approximation T , are bounded by

√ 1 + A/ √ 1 -B and the Lipschitz constant of its inverse T -1 by √ 1 + B/ √ 1 -A. • For all i, j, a, we have |Q a ij | ≤ 3d 2 ∂g max,x (1 + C) √ 1 + B √ 1 -A + d 3 ∂g max,y (1 + A)(1 + C) 1 -B .
We do not assume we are using Riemann normal coordinates in the statement of the lemma, but the notation is chosen to be compatible with this. The metric is close to Euclidean metric whose metric (δ ij ) is often denoted by the identity matrix, but formally is a (covariant) two tensor. The derivatives of the metric (g max,x , g max,y ) have been bounded in Riemann normal coordinate neighbourhoods in Corollary 4.3.

Proof of Lemma 4.7 We write G for the matrix g ij (0), and we assume that the eigenvalues λ i , are not far from 1, that is |λ i -1| ≤ A for all i and some A ≥ 0. Similarly, we write G for g ab , and assume that its eigenvalues λi are bounded by | λi -1| ≤ B for some B ≥ 0. We'll also write G = o t A D A o A with o A the orthogonal matrix that diagonalizes G and D A the diagonal matrix with the eigenvalues of G on the diagonal, that is diag(λ i ). We let S A denote the matrix with square roots √ λ i on the diagonal, that is S A = diag( √ λ i ). And similarly, G = o t B D B o B and S B = diag( λi ). Now [START_REF] Boissonnat | The reach, metric distortion, geodesic convexity and the variation of tangent spaces[END_REF] gives We shall now consider the quadratic term. We start with

G = T t GT o t A S A S A o A = T t o t B S B S B o B T o t A S t A S A o A = T t o t B S t B S B o B T I = S -t A o A T t o t B S t B S B o B T o t A S -1 A I = o t o, that is S B o B T o t A S -1 A = o, with o an orthogonal transformation. This in turn implies that T = o t B S -1 B oS A o A is
∂ k g ij = a,b,c ∂cg ab T c k T a i T b j + a,b g ab (Q a ik T b j + T a i Q b jk ).
Reshuffling and permuting the indices and changing the sign for the last equation gives

∂ k g ij - a,b,c ∂cg ab T c k T a i T b j = a,b g ab Q a ik T b j + a,b g ab T a i Q b jk ∂ i g jk - a,b,c ∂cg ab T c i T a j T b k = a,b g ab Q a ji T b k + a,b g ab T a j Q b ki -∂ j g ki + a,b,c ∂cg ab T c j T a k T b i = - a,b g ab Q a kj T b i - a,b g ab T a k Q b ij .
Adding the terms yields

∂ k g ij + ∂ i g jk -∂ j g ki - a,b,c ∂cg ab T c k T a i T b j - a,b,c ∂cg ab T c i T a j T b k + a,b,c ∂cg ab T c j T a k T b i = a,b g ab Q a ik T b j + a,b g ab T a i Q b jk + a,b g ab Q a ji T b k + a,b g ab T a j Q b ki - a,b g ab Q a kj T b i - a,b g ab T a k Q b ij = 2 a,b g ab Q a ik T b j ,
and thus

j,e ∂ k g ij + ∂ i g jk -∂ j g ki (T -1
) j e g ef The reach of subsets of manifolds

- a,c,e ∂cgaeT c k T a i g ef - b,c,e ∂cg eb T c i T b k g ef + a,b,e ∂eg ab T a k T b i g ef = j,e ∂ k g ij + ∂ i g jk -∂ j g ki - a,b,c ∂cg ab T c k T a i T b j - a,b,c ∂cg ab T c i T a j T b k + a,b,c ∂cg ab T c j T a k T b i (T -1 ) j e g ef =2 a,b,e,j g ab Q a ik T b j (T -1
) j e g ef =Q f ik , The idea now is the following: If we assume that the left hand side of the previous equation is close to zero (this is in line with Corollary 4.3 because we assume that the derivatives of the metric are not too large if the neighbourhood is not too big), g ab is close to δ ab , and T b j is close to a rotation, all entries of Q a ij have to be close to zero too.

Let us now assume that for all k, i, j we have that

|∂ k g ij | ≤ ∂gmax,x,
and for all a, b, c, that

|∂ag bc | ≤ ∂gmax,y.
We'll also assume that entries of g ef are bounded in absolute value by 1 + C. We will use that for a tensor Uµν , with |Uµν | ≤ Umax, and the coordinates of vectors v µ , w µ are bounded by |v µ | ≤ vmax and |w µ | ≤ wmax, we have that

µ,ν Tµν v µ w ν ≤ µ,ν Tmaxvmaxwmax = d 2 Tmaxvmaxwmax, (6)
as well as its obvious generalization. Thanks to the triangle inequality we now have

|Q f ik | = j,e ∂ k g ij + ∂ i g jk -∂ j g ki (T -1 ) j e g ef - a,c,e ∂cgaeT c k T a i g ef - b,c,e ∂cg eb T c i T b k g ef + a,b,e ∂eg ab T a k T b i g ef ≤ j,e ∂ k g ij + ∂ i g jk -∂ j g ki (T -1 ) j e g ef + a,c,e ∂cgaeT c k T a i g ef + b,c,e ∂cg eb T c i T b k g ef + a,b,e ∂eg ab T a k T b i g ef ≤3d 2 ∂gmax,x (1 + C) √ 1 + B √ 1 -A + d 3 ∂gmax,y (1 + A)(1 + C) 1 -B □ Remark 4.8
In the previous lemma we have given a bound on the metric distortion (on the coordinates) of the coordinate transformation from x to y, as well as on the Lipschitz constant of the derivative of the transformation. In the formulation of the lemma we have chosen the coordinate systems such that the coordinates map the origin to the origin to keep the expressions short and relatively simple. However we could have chosen the map to go from an arbitrary x 0 to y 0 . In other words, this choice of coordinates is not canonical, unlike the origin in Riemann normal coordinate systems which is canonical.

Using Federer's estimates: from bounds on the coordinate transformation to bounds on the reach

In this section, we are finally able to give bounds on the reach by applying Theorem 4.19 of Federer [START_REF] Federer | Curvature measures[END_REF]:

Theorem 4.9 (Federer) Let S be a subset of R d with rch(S) > t > 0, and s > 0.

If f : {x | d(x, S) < s} → R d is a C 2 diffeomorphism such that |D f | ≤ M |D f -1 | ≤ N |D 2 f | ≤ P,
where D denotes the derivative, then

rch( f (S)) ≥ min{sN -1 , (M t -1 + P ) -1 N -2 }.
We can now combine this result with the estimates of the previous sections. We want to investigate how an empty tangent ball to S transforms under the exponential map. Because a geodesic ball is also an Euclidean ball in the tangent space of its centre (lifted via the exponential map), this is equivalent to giving bounds on the reach of this ball under the map exp -1 p • exp c . The bounds on the reach under the map exp -1 p • exp c use almost all previous results in this paper: In particular the bounds on the metric and its derivatives are given in Lemma 4.1, and Corollaries 4.3, 4.4, and 4.6, while Lemma 4.7 tells us how to go from bounds on the metric to bounds on the coordinate transformation. Federer's result now gives us the reach after the transformation.

We can now give the two main results of the paper. The first gives a bound on the reach of a lifted (via the exponential map at a nearby point) geodesic sphere. The second uses this result to give a lower bound on the local feature size on the lifting of any set of positive reach.

|R σ µνλ | ≤ Rmax (2) |∇κR σ µνλ | ≤ R ∇ max . ( 3
)
Suppose that rp and rc are the radii of geodesic balls centred at p and c respectively such that Then the reach of exp -1 p (∂B(c, rc)) ⊂ TpM is lower bounded. Specifically,

rch(exp -1 p (∂B(c, rc))) ≥ min{rcN -1 , (M r -1 c + P ) -1 N -2 }, where M = √ 1 + A √ 1 -B N = √ 1 + B √ 1 -A P = 12∂gmax,x (1 + C) √ 1 + B √ 1 -A + 8∂gmax,y (1 + A)(1 + C) 1 -B , with A = dRmaxr 2 c B = dRmaxr 2 p . C = dRmaxr 2 p 1 -dRmaxr 2 p ,
and ∂gmax,x = ∂gmax(rc), ∂gmax,y = ∂gmax(rp), with

∂gmax(r) = 20Rmaxr + 10R ∇ max r 2 (1 -Rmaxd r 2 ) 3/2 .
We now also find that Theorem 4.11 Suppose that M satisfies the same conditions as in Theorem 4.10 and let S be a subset of M of positive reach rch(S). Let q ∈ S and let B(p, rp) be a geodesic ball, whose radius satisfies • 10 ρ < r p for some ρ ≤ rch(S), • d(q, p) <

(q) (Sp) ≥ L where L = min{rch(S)N -1 , (M rch(S) -1 + P ) -1 N -2 }.
Proof The proof is more complicated than first appears because for general sets of positive reach we cannot directly work with the (classical) normal space. 3 To derive a contradiction assume that for some L > ϵ > 0 there is a p ′ ∈ B E (q, ϵ) ⊂ TpM, such that π exp -1 p (S) (p ′ ) consists of multiple points. The latter statement is equivalent to: there is a L > r ′ > 0 such that B ′ ∩ Sp = ∂B ′ ∩ Sp contains more than one point, where B ′ = B E (p ′ , r ′ )⊂ TpM. Let q ′ be such a point and consider T q ′ ∂B ′ . We stress that here ∂B ′ is seen as a submanifold of TpM, which is (isometric to) a Euclidean space, and thus T q ′ ∂B ′ ⊂ TpM.

Because the manifold M is smooth (C 3 ) and we work in a neighbourhood that is smaller than the injectivity radius exp p is a smooth map whose differential maps T q ′ ∂B ′ ⊂ TpM to a subspace ST q ′ B of T exp p (q ′ ) M. Moreover, for the same reason, exp p ∂B ′ is a C 2 manifold and therefore has positive reach by Theorem 3.1. It also has an interior and an exterior by the Jordan-Brouwer theorem, see for example [START_REF] Guillemin | Differential Topology[END_REF] for a pedagogical proof of the Jordan-Brouwer theorem. We note that ST q ′ B is tangent to exp p ∂B ′ and the interior of exp p B ′ has an empty intersection with S. Write ν for the interior unit normal vector of exp p ∂B ′ and ST q ′ B at exp p (q ′ ).

Let γν (t) be the geodesic, parametrized by arc length, that starts at exp p (q ′ ) and goes in the direction of ν. Using Theorem 3.1 once more, we see that for sufficiently small ϵ ′ > 0 we have π exp p ∂B ′ (γν (t)) = π S (γν (t)), for all t ′ ≤ ϵ ′ . Thanks to Kleinjohann [33, Satz 3.2 and 3.3], we have that this minimizing geodesic of length ϵ ′ can be extended to a geodesic of length rch(S), which is larger than ρ by assumption. Therefore, for each rc < rch(S), the geodesic ball Bc = B(γν (rc), rc) intersects S only in exp p (q ′ ) and its boundary is tangent to ST q ′ B at exp p (q ′ ). We note that exp -1 p (∂Bc) is tangent to T q ′ ∂B ′ in the point q ′ . We can now use Theorem 4.10 to see that the reach of exp -1 p (∂Bc) is lower bounded by Lc := min{rcN -1 , (M r -1 c + P ) -1 N -2 }, using the notation as defined in Theorem 4.10. If r ′ < Lc for all rc < ρ, then we have reached a contradiction, because B ′ ∩ Sp = ∂B ′ ∩ Sp was supposed to contain more than one point.

We note that r ′ < Lc for all rc < ρ is equivalent to r ′ < L. The result now follows. □ Remark 4. [START_REF] Cheng | Delaunay Mesh Generation[END_REF] We have formulated Theorem 4.11 in terms of the local feature size, because we did not want to exclude subsets whose size is larger than the injectivity radius and therefore would not fit in its entirety in a single coordinate chart given by the exponential map.

Because we know that a topologically embedded submanifold of R d has positive reach if and only if it is C 1,1 embedded, see [START_REF] Lieutier | Embedded topological manifolds of positive reach with boundary of positive reach are C[END_REF], we immediately have the following corollary: As mentioned in the introduction of Section 4 this result was known to Lytchak [START_REF] Lytchak | On the geometry of subsets of positive reach[END_REF][START_REF] Lytchak | Almost convex subsets[END_REF], but his work heavily relies on the theory of CAT(k) spaces, while [START_REF] Lieutier | Embedded topological manifolds of positive reach with boundary of positive reach are C[END_REF] is more elementary.

We further notice Nor(p, S) := Nor(p, exp -1 p S). These spaces are convex cones (by definition), which is not obvious from [START_REF] Kleinjohann | Nächste Punkte in der Riemannschen Geometrie[END_REF].

Future work

As we mentioned in the introduction the main motivation of this work (at least for the authors) is the triangulation of stratified Riemannian manifolds. To make the importance of the bounds of the paper clear in this context we will discuss the approach for a Riemannian manifold with a submanifold inside (the simplest example of a stratified Riemannian manifold).

We are able to Delaunay triangulate the Riemannian manifold using intrinsic simplices [START_REF] Boissonnat | Delaunay Triangulation of Manifolds[END_REF]. If the intersection of the submanifold with every top dimensional intrinsic simplex in the Delaunay triangulation is nice, i.e. if the intersection after lifting the tangent space of a nearby point is a slightly deformed polytope, we can triangulate relatively straightforwardly using barycentric subdivision, in generalization of Whitney's approach [14,[START_REF] Whitney | Geometric Integration Theory[END_REF].

In the Euclidean setting one established this property by perturbing the triangulation in such a way that the (sub)manifold stays sufficiently far away from the d -n -1-skeleton of the ambient triangulation, where sufficiently far is a bound in terms of the reach. In the Riemannian setting one would like to apply (a variant of) this method after (locally) lifting to the tangent space (at a nearby point) of the Riemannian manifold. For this one needs two supporting results, namely to bound the reach of the lifted submanifold (which we discussed in this paper) and to understand the geometry of the faces of the Riemannian simplices in the ambient triangulation.

research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 754411 and the Austrian science fund (FWF) M-3073.

Lemma 4 . 5 Corollary 4 . 6

 4546 Let E be a d × d-matrix, and G = I + E, with I the identity matrix. If λ is an eigenvalue of G, then |λ -1| ≤ ∥E∥ 2 . Using (4) again now gives, If d M (p, x) ≤ r, with Rmaxr 2 ≤ π 2 4 and r ≤ ι M 2 , any eigenvalue λ of g ij (x) now satisfies |λ -1| ≤ dRmaxr 2 , using Riemann normal coordinates around p.

  close to an orthogonal transformation if A and B are close to zero. The Lipschitz constant of a composition of function is the product of the Lipschitz constants and thus the Lipschitz constant of T is bounded by (1 -A)(1 -B) and (1 + A)(1 + B) respectively. Because we have that for any vector |v µ | ≤ |v|, where the first | • | should be read as an absolute value and the second as the norm, and A ij = e t i Ae j , where the e i denote basis vectors, the entries of T are bounded by from above by √ 1 + A/ √ 1 -B. By the same argument we have that the entries of T -1 are bounded by √ 1 + B/ √ 1 -A.

Theorem 4 .

 4 10 Let M be a smooth d-dimensional Riemannian manifold whose curvatures are bounded as follows:

Corollary 4 .

 4 13 A topologically embedded submanifold S of a C 3 manifolds M has positive reach if and only if S is C 1,1 embedded.

Remark 4 .

 4 14 In [3, Definitions 4.3 and 4.4 and Theorem 4.8] Federer shows that the generalized tangent space Tan and generalized normal space Nor of a set of positive reach or positive local feature size are convex cones. Theorem 4.11 says that the lifting (via exp p ) set of positive reach has positive local feature size. Combining these two observations makes the following definition sensible Tan(p, S) := Tan(p, exp -1 p S)

  we have the following

Corollary 4.4 If d M (p, x) ≤ r, with dRmaxr 2 ≤ 1 and r ≤ ι M 2 , then |g ij (x) -δ ij | ≤ dRmaxr 2 1 -dRmaxr 2 ,

using Riemann normal coordinates around p.

  • B(c, 2r c ) ⊂ B(p, r p ), where we have made the centres and radii explicit, • √ dR max r 2 p ≤ 1 and r p ≤ ι M 2 .

By boundary we mean the set minus its interior for a closed set, and the closure of the set minus the interior of the closure for an open set.

The reach of subsets of manifolds

Even in Euclidean space, the normal spaces of sets of positive reach are not necessarily vector spaces. We'll see in Remark

4.14, that normal cones in the sense of Federer[START_REF] Federer | Curvature measures[END_REF] are well defined, also for subsets of Riemannian manifolds.

Acknowledgements

We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu for encouragement. We further acknowledge the anonymous reviewers whose comments helped improve the exposition.

Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions).

The first author is further supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. The second author is supported by the European Union's Horizon 2020

Conflicts of interest/Competing interests

There are no conflicts of interest.


Alexandre Weppe
email: alexandre.weppe@ensma.fr

Didier SAURY Florian Moreau

Experimental study of the flow dynamics in a confined cubic

Keywords: Etude expérimentale, Convection naturelle Experimental study, Natural convection. Nomenclature

un écoulement dans une cavité cubique comprenant un bloc partiellement chauffé.

viscosité dynamique à T 0 , kg.m -1 .s -1 α diffusivité thermique à T 0 , m 2 .s -1 β coefficient de dilatation thermique volumétrique à T 0 , K -1 ν viscosité cinématique à T 0 , m 2 .s -1 Indices c relatif à la paroi chaude f relatif à la paroi froide as relatif à l'écoulement ascendant des relatif à l'écoulement descendant

Introduction

Les écoulements turbulents à effets de flottabilité dominants sont présents dans de nombreuses applications industrielles, notamment dans les secteurs du nucléaire et de l'automobile. On peut par exemple citer la problématique du refroidissement d'un compartiment moteur qui est un point essentiel dans le dimensionnement d'un véhicule. Suite à un arrêt brutal du moteur après une forte sollicitation, l'intégrité de celui-ci doit être préservée alors même qu'il n'est plus refroidi par un écoulement externe forcé. Pour de telles situations, la convection, souvent en régime turbulent, permet d'assurer le refroidissement.

Le projet, au sein duquel s'inscrit cette étude expérimentale, a pour ambition de résoudre les problèmes rencontrés par les partenaires industriels lors de simulations d'écoulements turbulents avec effets de flottabilité dominants en espace confiné et aussi d'améliorer la compréhension des phénomènes physiques observés pour ce type d'écoulement.

Notre démarche a consisté à définir une configuration de référence permettant d'étudier un écoulement en espace confiné représentatif des régimes rencontrés dans le domaine automobile. Le compartiment moteur a été simplifié en une cavité cubique comportant en son sein un bloc cubique partiellement chauffé. Plusieurs études, en majorité numériques, traitent de l'influence d'un obstacle dans une cavité. Ainsi certaines études démontrent, à l'intérieur d'une cavité cubique, l'impact de la taille et de la conductivité d'un bloc non-chauffé [START_REF] House | Effect of a Centered Conducting Body on Natural Convection Heat Transfer in an Enclosure[END_REF][START_REF] Bhave | Natural convection heat transfer enhancement using adiabatic block : Optimal block size and Prandtl number effect[END_REF] quand d'autres s'intéressent plus particulièrement au rôle d'un bloc chauffant sur les transferts conjugués conduction/convection. [START_REF] Liu Et Pan-Thien | A complete conjugate conduction convection and radiation problem[END_REF][START_REF] Ha | A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body[END_REF]. Comme il sera montré dans la section 2 et avec la figure 2, la zone en orange qui est la zone étudiée dans nos travaux a des similarités avec les écoulements rencontrés dans une cavité différentiellement chauffée. Les écoulements turbulents générés à l'intérieur d'une cavité différentiellement chauffée ont déjà été étudiés expérimentalement [START_REF] Saury | Natural convection in an air-filled cavity : Experimental results at large Rayleigh numbers[END_REF][START_REF] Salat | Experimental and numerical investigation of turbulent natural convection in a large airfilled cavity[END_REF][START_REF] Betts | Experiments on turbulent natural convection in an enclosed tall cavity[END_REF]. Cependant, contrairement à ces études, la "cavité différentiellement chauffée" étudiée ici est ouverte.

Dans un premier temps, nous allons présenter le dispositif expérimental ainsi que la métrologie utilisée pour nos mesures puis nous analyserons les résultats obtenus avant de conclure sur cette étude préliminaire.

Dispositif expérimental et métrologie 2.1. Dispositif expérimental

La cavité utilisée dans cette étude est une cavité cubique de côté L = 1 m dans laquelle est placé un bloc chauffant de côté L bloc = 0, 8 m (voir Fig. 1 et 2).

Deux échangeurs à eau en aluminium (conductivité λ = 174 W.m -1 .K -1 ) imposent une température T f = 292 K aux parois latérales de la cavité cubique au moyen de deux bains thermostatés Fig. 1. Les parois horizontales haute et basse de la cavité sont en polystyrène extrudé (conductivité λ = 0,035 W.m -1 .K -1 ) et sont donc considérées comme adiabatiques. Elle sont de plus recouvertes d'une feuille de Mylar faiblement émissive ( = 0,044). Les parois avant et arrière de la cavité cubique sont quant à elles en verre d'épaisseur 6 mm permettant ainsi de visualiser l'écoulement. Des blocs de polystyrène extrudé sont maintenus devant ces parois afin de conserver leur caractère adiabatique tout en laissant la possibilité de libérer des espaces pour la visualisation.

Le bloc cubique à l'intérieur de la cavité est maintenu par quatre supports en polymère fixés sur des tiges filetées. Les tiges sont entourées de tubes en PVC et traversent la paroi basse pour

Conditions limites

Chacune des faces avant, arrière, haute et basse du bloc sont instrumentées avec 9 thermocouples. 14 thermocouples sont fixés sur la face intérieure de chacune des plaques d'aluminium du bloc cubique. Enfin, 10 thermocouples sont positionnés à l'intérieur du bloc. L'ensemble de ces 74 thermocouples ont permis un suivi précis des conditions de températures autour et à l'intérieur du bloc et ont permis de vérifier l'homogénéité en température de la paroi chaude (écart-type=0,6K).

Mesures de vitesses

Les mesures de vitesses sont effectuées par PIV (Particle Image Velocimetry). Un générateur de fumée produit le traceur de l'écoulement (particules d'huile de paraffine : ρ = 856 kg.m -3 , diamètre = 5 µm) et la nappe laser est créée par un laser pulsé Nd :YAG Litron. Le nombre de Stokes associé Stk 7×10 - 4 1 indique que les particules suivent précisément l'écoulement.

Les particules sont visualisées à l'aide d'une caméra Tokina F100 de résolution 2320 × 1720 pixels 2 . La taille d'un pixel est de 7 µm et la taille du champ est fixée à 97 × 72 mm 2 . La largeur de l'image correspond à la largeur l de l'espace entre la paroi chaude à T c et la paroi froide à T f . 6000 doublets d'images sont acquis à une fréquence de 96 Hz pour chaque mesure et la durée entre deux images varie selon la zone visualisée (entre 1000 et 5000 µs). 6 zones sont étudiées le long de notre cavité (Fig. 2). Pour chaque zone, les vitesses verticales et horizontales sont analysées par la suite. Au sein d'une même zone, les faibles vitesses observées au centre de la cavité nécessitent de prendre les premières images de chaque doublet. De nouvelles paires d'images adaptées à de petits déplacements sont ainsi créées.

Le traitement d'images est effectué avec l'algorithme "Adaptative PIV" de Dantec®. Cet algorithme adapte la taille des fenêtres d'interrogations (de 64x64 à 32x32) pour gagner en précision dans les zones où la densité de particules est suffisante. Un overlapping variant entre 75% et 50% permet de suivre les changements de tailles des fenêtres.

Recollement des données "basses" et "hautes" vitesses

Les champs moyens et RMS sont associés soit aux faibles vitesses soit aux vitesses plus importantes. Ils nécessitent donc d'être recoupés pour chaque zone.

Un critère seuil est choisi afin de délimiter, pour une zone, le champ des "hautes" vitesses et le champ des "basses" vitesses. Ce critère est déterminé grâce à la norme de la vitesse en pixels de déplacements des particules de la zone concernée. Si cette norme est inférieure à un pixel de déplacement, le champ des basses vitesses est choisi. En effet, en dessous du pixel de déplacement, les valeurs des vitesses associées aux champs des hautes vitesses sont calculées par l'algorithme et non plus mesurées. Afin d'assurer une transition continue entre les champs de vitesses hautes et basses, on choisit d'utiliser une fonction de transition classique comprenant une tangente hyperbolique :

V = 1 2 • 1 + tanh x d -x p • BV + 1 2 • 1 + tanh x -x d p • HV (1)
avec comme variables, x d : valeur frontière (critère seuil) ; p : facteur de largeur ; HV : hautes vitesses ; BV : basses vitesses.

La valeur frontière x d correspond à la position x à laquelle on choisit d'utiliser soit le champ des hautes vitesses soit le champ des basses vitesses. Cette valeur frontière est obtenue à partir des données expérimentales et varie selon la zone étudiée : x d = f (Y ). La valeur de p est choisie de fac ¸on à minimiser le "saut" de vitesses observé entre le champ des hautes vitesses et le champ des basses vitesses mais modifie un plus grand nombre de points autour de la valeur frontière x d . La valeur de p est fixée à p = 5mm afin de limiter ce "saut" sans modifier de manière importante les points avoisinants.

Résultats

Les mesures ont été effectuées dans le plan médian. Des profils de vitesses moyennes sont tracés pour différentes côtes choisies équitablement le long de la cavité (Fig. 3) avec l'origine Y = 0 mm prise sur la partie supérieure de la paroi basse de la cavité. La position de ces côtes est schématisée Fig 2 . L'écoulement est ascendant le long de la paroi à température T c et le développement d'une couche limite est observé. La couche limite s'épaissit de la zone Y = 228 mm à la zone Y = 775 mm. Elle devient nulle dès lors que l'écoulement ne rencontre pas ou plus la paroi (Y = 6 mm et Y = 957 mm). Une couche limite suivant l'écoulement descendant est également observée le long de la paroi à température T f . La couche limite descendante est moins épaisse que la couche limite ascendante. De plus, le maximum de vitesse en valeur absolue pour la couche limite descendante est environ quatre fois inférieur à celui de la couche limite ascendante (pour Y = 593 mm : V max/as = 0,360m.s -1 , V max/des = 0,085m.s -1 ). Cette différence s'explique par le fait qu'une partie de l'écoulement ascendant se dirige vers le canal haut comme le montre le profil de vitesses horizontales moyennes pour Y = 957 mm (Fig4).

Pour les zones présentant une couche limite sur la paroi froide et la paroi chaude, les vitesses moyennes horizontales et verticales situées dans la zone centrale de la cavité sont proches de 0 m.s -1 . Cette zone peut donc être considérée comme au repos en moyenne. Néanmoins, à mesure que l'on s'élève dans la cavité, les valeurs RMS des vitesses horizontales (correspondant à l'enveloppe autour des courbes de vitesse moyenne et représentées par des barres verticales de hauteur 2σ) proches de la paroi froide et dans la zone centrale augmentent significativement. Elles traduisent des perturbations de plus en plus importantes au sein de l'écoulement. La couche limite ascendante est moins affectée par ces perturbations. La vitesse, très majoritairement verticale, y est élevée et ne subit que très peu l'influence de l'écoulement situé dans la zone centrale.

Les zones Y = 6 mm et Y = 957 mm présentent des caractéristiques particulières. Elles sont dépourvues de limite à leur droite du fait de l'absence de paroi. Ainsi, la zone Y = 6 mm voit une partie de l'écoulement provenant de la couche limite descendante se diriger vers le canal bas. Cela se traduit par des vitesses horizontales moyennes positives. A l'inverse, et comme l'on se trouve très proche de la paroi basse, les vitesses verticales moyennes sont nulles. Pour la zone Y = 957 mm, l'écoulement qui se dirige vers le canal haut possède cette fois-ci une composante verticale moyenne et une composante horizontale moyenne non négligeables. La cote choisie est représentative du "virage" amorcé par l'écoulement.

Conclusion

Cette étude préliminaire a permis de caractériser la topologie globale de l'écoulement au sein de la cavité. Des mesures de vitesses ont été effectuées dans le plan médian. Un écoulement de couches limites avec vitesses nulles au centre de la cavité a pu être constaté. Des perturbations dans la zone centrale et proche de la paroi froide ont été mises en évidence. la direction globale de l'écoulement aux extrémités haute et basse de la cavité a également été étudiée. Une prochaine étape sera de pouvoir étudier les transferts de chaleur au sein de la cavité.

L

  longueur de la cavité, m l largeur de la zone étudiée, m H bloc hauteur du bloc chauffant, m Ra Hbloc nombre de Rayleigh basé sur H bloc , Ra Hbloc = gβ(Tc-Tf )H 3 bloc να d diamètre d'une particule, m Stk nombre de Stokes, Stk = ρd 2 vref 18µl T température, K T 0 température moyenne, T 0 = 1 2 (T c + T f ), K v ref vitesse de référence de l'écoulement, v ref = α Hbloc Ra Hbloc , m.s -1 X, Y, Z coordonnées physiques, m Symboles grecs σ écart-type ρ masse volumique à T 0 , kg.m -3 µ

Figure 1 :

 1 Figure 1 : Vue 3D de la cavité

Figure 2 :

 2 Figure 2 : Vue du plan à mi-profondeur de la cavité (profils de vitesses (en pointillés rouges))

Figure 3 :Figure 4 :

 34 Figure 3 : Profils horizontaux de vitesse verticale moyenne, V, à différentes altitudes dans le canal chauffé. Les barres verticales correspondent aux vitesses RMS (hauteur 2σ).

Remerciements

Les auteurs remercient l'ANR pour le financement du projet MONACO 2025 ; les programmes CPER (2015-2020) et FEDER (2014-2020) ; ainsi que le personnel d'appui qui a participé à la mise en place de ce dispositif expérimental, en particulier H. Arlaud, C. Fuentes, J.-C. Fraudeau et N. Papin.