Colin Éric

De Verdière

Thomas Magnard

An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes

Keywords: 2012 ACM Subject Classification Theory of computation → Computational geometry, Mathematics of computing → Graph algorithms, Mathematics of computing → Graphs Keywords and phrases computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability Digital Object Identifier 10.4230/LIPIcs.ESA.2021.32

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface.

It is known that the problem admits an algorithm with running time f (c)n O(c) , where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f (c)n 3 -time algorithm. We also provide a randomized algorithm with expected running time 2 c O(1)

n O(1) . Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.

Introduction

An embedding of a graph G into a host topological space X is a crossing-free topological drawing of G into X. The use and computation of graph embeddings is central in the communities of computational topology, topological graph theory, and graph drawing. A landmark result is the algorithm of Hopcroft and Tarjan [START_REF] Hopcroft | Efficient planarity testing[END_REF], which allows to decide whether a given graph is planar (has an embedding into the plane) in linear time. Related results include more planarity testing algorithms [START_REF] Patrignani | Planarity testing and embedding[END_REF], algorithms for embedding graphs on surfaces [START_REF] Mohar | A linear time algorithm for embedding graphs in an arbitrary surface[END_REF][START_REF] Kawarabayashi | A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width[END_REF] and for computing book embeddings [START_REF] Malitz | Genus g graphs have pagenumber O( √ g)[END_REF], Hanani-Tutte theorems [START_REF] Schaefer | Toward a theory of planarity: Hanani-Tutte and planarity variants[END_REF], and the theory of crossing numbers and planarization [START_REF] Buchheim | Crossings and planarization[END_REF].

In this paper, we describe algorithms for deciding the embeddability of graphs into topological spaces that are, in a sense, as general as possible: two-dimensional simplicial complexes (or 2-complexes for brevity), which are made from vertices, edges, and triangles glued together. (We remark that every graph is embeddable in R 3 , so considering higherdimensional simplicial complexes is irrelevant.) In a previous article, jointly written with Mohar [START_REF] Éric Colin De Verdière | Embedding graphs into twodimensional simplicial complexes[END_REF], we proved that, given a graph G and a 2-complex C , one can decide whether G embeds into C in polynomial time for fixed C ; but the algorithm has running time f (c) • n O(c) , where n and c are the respective sizes of G and C . Using a very different strategy, we describe algorithms for this problem, proving that it is fixed-parameter tractable in the complexity of the input complex: ▶ Theorem 1.1. One can solve the embeddability problem of graphs into 2-dimensional simplicial complexes in deterministic f (c)n 3 time or in expected time 2 c O (1) n O (1) , where c is the number of simplices of the input 2-complex, n is the number of vertices and edges of the input graph, and f is some computable function of c.

2-complexes are much more general than surfaces, and tools that are suitable for studying embeddability of graphs on surfaces do not generalize. For example, the set of graphs embeddable on a given 2-complex is not closed under minor, which makes many tools for dealing with graphs on surfaces unsuitable for 2-complexes. Moreover, the complexity of some topological problems increase drastically when we consider 2-complexes instead of surfaces, e.g., deciding homeomorphism [START_REF] Colm Ó Dúnlaing | Homeomorphism of 2-complexes is equivalent to graph isomorphism[END_REF], or deciding the contractibility of curves [START_REF] Tamal | Transforming curves on surfaces[END_REF][START_REF] Lazarus | On the homotopy test on surfaces[END_REF][START_REF] Erickson | Transforming curves on surfaces redux[END_REF]. Some other topological problems, such as the existence of a drawing a graph with at most k crossings in the plane or in a surface [START_REF] Kawarabayashi | Computing crossing number in linear time[END_REF], can be recast as deciding whether the graph embeds on a certain 2-complex. For more detailed motivations, see [START_REF] Éric Colin De Verdière | Embedding graphs into twodimensional simplicial complexes[END_REF]Introduction].

Comparison with previous works on surfaces

Since every surface is homeomorphic to a 2-complex, our problem has been largely considered in the special case where the input 2-complex is (homeomorphic to) a surface. That restricted problem is NP-hard [START_REF] Thomassen | The graph genus problem is NP-complete[END_REF], but several algorithms that are fixed-parameter tractable in the genus have been given, which we review now.

Mohar [START_REF] Mohar | A linear time algorithm for embedding graphs in an arbitrary surface[END_REF] has given an algorithm for this purpose that takes linear time in the input graph, for every fixed surface. This algorithm is very technical and relies on several other articles. The dependence on the genus is not made explicit, but seems to be doubly exponential [START_REF] Kawarabayashi | A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width[END_REF]. Kawarabayashi et al., in an extended abstract [START_REF] Kawarabayashi | A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width[END_REF], have given a simpler linear-time algorithm for this problem, but not all details are presented, which makes the approach hard to check [15, p. 3657, footnote].

General graph minor theory provides an algorithm for the same purpose. The graph minor theorem by Robertson and Seymour [START_REF] Robertson | Graph minors. XX. Wagner's conjecture[END_REF] implies that, for every fixed surface S , there is a finite list of graphs O S such that a graph G can be embedded on S if and only if G does not contain any graph in O S as a minor. Moreover, there is an algorithm that given any surface S (specified by its genus and orientability) outputs the list O S [START_REF] Adler | Computing excluded minors[END_REF], and there is an algorithm to decide whether a graph M is a minor of another graph G running, for fixed M , in time cubic in the size of G [START_REF] Robertson | Graph minors. XIII. The disjoint paths problem[END_REF][7, Theorem 6.12]. These considerations thus lead to an algorithm to decide embeddability of a graph on a surface that runs, if the input surface is fixed, in cubic time in the size of the input graph.

Finally, in the same vein, Kociumaka and Pilipczuk [START_REF] Kociumaka | Deleting vertices to graphs of bounded genus[END_REF] have studied the following more general problem than the embeddability problem of graphs on surfaces: Given a surface S , a graph G, and an integer k ≥ 0, is it possible to remove a set U of at most k vertices from G so that G -U is embeddable on S ? They provide an algorithm that is fixed-parameter tractable in k and the genus of S , where the dependence on the genus is unspecified. In particular, as a special case, they decide the embeddability of a graph on a surface; however, they use one of the previous algorithms [START_REF] Mohar | A linear time algorithm for embedding graphs in an arbitrary surface[END_REF][START_REF] Kawarabayashi | A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width[END_REF] as a subroutine. The problem that we study, the embeddability of graphs on 2-complexes, is independent from the problem studied by Kociumaka and Pilipczuk, in the sense that there is, a priori, no obvious reduction from one problem to the other. However, we will reuse some ingredients from that paper.

Our algorithms, restricted to the case where we want to embed graphs on surfaces, are not as efficient as the existing algorithms mentioned above. Indeed, the deterministic one runs in cubic time in the size of the input graph (for a fixed complex); the dependence on the size of the complex is not made explicit, because the algorithm uses, as a subroutine, an algorithm to compute grid minors [START_REF] Robertson | Graph minors. XIII. The disjoint paths problem[END_REF]. The second algorithm is randomized, because it uses an algorithmic version of the excluded grid theorem [START_REF] Chekuri | Polynomial bounds for the grid-minor theorem[END_REF] that uses randomness; for every fixed surface, it runs in expected time that is a polynomial of fixed (but large) degree in the size of the input graph. However, our algorithms are independent from the existing algorithms for embedding graphs on surfaces; the only elaborated tool that we use is an algorithm to compute grid minors [START_REF] Chekuri | Polynomial bounds for the grid-minor theorem[END_REF][START_REF] Robertson | Graph minors. XIII. The disjoint paths problem[END_REF].

Overview and structure of the paper

We use a standard strategy in graph algorithms and parameterized complexity (see, e.g., the book by Cygan et al. [START_REF] Cygan | Parameterized algorithms[END_REF]Chapter 7]): we show by dynamic programming that the problem can be solved efficiently for graphs of bounded branchwidth, and then, using an irrelevant vertex method, we prove that one can assume without loss of generality that the input graph G has branchwidth bounded by a polynomial in the size of the input 2-complex. In the context of surface-embedded graphs, this paradigm has been used in the extended abstract by Kawarabayashi et al. [START_REF] Kawarabayashi | A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width[END_REF] and in the article by Kociumaka and Pilipczuk [START_REF] Kociumaka | Deleting vertices to graphs of bounded genus[END_REF]; our algorithm takes inspiration from the former, for the idea of the dynamic programming algorithm, and from the latter, for some arguments in the irrelevant vertex method. However, handling 2-complexes requires significantly more effort. More precisely, Theorem 1.1 follows immediately from the following two theorems.

▶ Theorem 1.2 (algorithm for bounded branchwidth). One can solve the embeddability problem of graphs into two-dimensional simplicial complexes in time (c + w) O(c+w) n, where c is the number of simplices of the input 2-complex, and where n and w are the number of vertices and edges and the branchwidth of the input graph, respectively. ▶ Theorem 1.3 (algorithm to reduce branchwidth). Let C be a 2-complex with c simplices, and G a graph with n vertices and edges. We can correctly report that G is embeddable on C , or correctly report that G is not embeddable on C , or compute a subgraph H of G, of branchwidth polynomial in the number of simplices of C , such that G embeds on C if and only if H does: in deterministic time f (c) • n 3 for some computable function f , or in expected polynomial time.

We now present the structure of the paper, indicating which techniques are used. We also emphasize which components would be simpler if we were just aiming for an algorithm for embedding graphs on surfaces.

We introduce some standard notions in Section 2. Then, in Section 3, we show that we can make some simple assumptions on the input, and present data structures for representing 2-complexes and graphs embedded on them. If we restrict ourselves to the case where the input 2-complex is homeomorphic to a surface, we essentially consider combinatorial maps of graphs on surfaces, except that the graphs need not be cellularly embedded. The case of 2-complexes is largely more involved.

In Section 4, we show that if our input graph G has an embedding into our input 2complex C , then there exists an embedding of G on C that is sparse with respect to a branch decomposition of G. This means that each subgraph of G induced by the leaves E S A 2 0 2 1

32:4

An FPT Algorithm for the Emebddability of Graphs into 2-Complexes of any subtree of the branch decomposition can be separated from the rest of G using a graph embedded on C , called partitioning graph, of small complexity. We find that this new structural result, even in the surface case, is interesting and can prove useful in other contexts. If the target space were a surface, we could assume that G is 3-connected and has no loop or multiple edges, which would imply (still with some work) that any embedding of G would be sparse, but again the fact that we consider 2-complexes requires additional work.

In Section 5, we present the dynamic programming algorithm, which either determines the existence of an embedding of G on C , or shows that no sparse embedding of G on C exists (and thus no embedding at all, by the previous paragraph). The idea is to use bottom-up dynamic programming and to consider all regions of the 2-complex in which the subgraph of G (induced by a subtree of the branch decomposition) can be embedded. The complexity depends exponentially on the branchwidth of G.

The previous arguments, most notably in Section 4, implicitly assumed that, if G has an embedding into C , it has a proper and cellular embedding, in particular, in which the faces are homeomorphic to disks. In Section 6, we show that we can assume this property. Essentially, we build all 2-complexes "smaller" than C , such that G embeds on C if and only if it embeds into (at least) one of these 2-complexes, and moreover if it is the case, it has an embedding into (at least) one of these 2-complexes that is proper and cellular. If C were an orientable surface, we would just consider the surfaces of lower genus; but here a more sophisticated approach is needed.

The above ingredients allow to prove Theorem 1.2 (Section 7).

In Section 8, we show, using an irrelevant vertex method, that we can assume that G has branchwidth polynomial in the size of C (Theorem 1.3).

Preliminaries

Graphs and branch decompositions

In this paper, graphs may have loops and multiple edges unless noted otherwise. Let G be a graph; as usual, we denote by V (G) and E(G) the sets of vertices and edges of G. A (rooted) branch decomposition of G is a rooted tree B in which: every node has degree either one (it is a leaf ) or three (it is an internal node), the root is a leaf, each non-root leaf is labelled with an edge of G, and this labelling induces a bijection. The vertices and edges of B are called nodes and arcs, respectively. Each arc α of B splits the tree B into two subtrees B 1 and B 2 ; if, for i = 1, 2, we denote by E i the set of labels appearing in T i , we see that α naturally induces a partition (E 1 , E 2 ) of the set of edges of G (if α is the arc incident to the root, then one part of the partition is empty). The middle set associated to α is the set of vertices of G which are the endpoints of at least one edge in E 1 and at least one edge in E 2 . The width of B is the maximum size of a middle set associated to an arc of B. The branchwidth of G is the minimum width of its (rooted) branch decompositions.

The usual definition of a branch decomposition is identical, except that the tree is unrooted, and thus the leaves are in bijection with the edges of G. Our definition turns out to be more convenient to use in the dynamic program. The difference is cosmetic, as one can transform one kind of branch decomposition into the other easily while preserving the width.

Surfaces

We will assume some familiarity with surface topology; see, e.g., [START_REF] Mohar | Graphs on surfaces[END_REF][START_REF] Stillwell | Classical topology and combinatorial group theory[END_REF][START_REF] Colin De Verdière | Computational topology of graphs on surfaces[END_REF]

closed half disk {(x, y) ∈ R 2 | y ≥ 0, x 2 + y 2 < 1}.

2-complexes

A 2-complex (or two-dimensional simplicial complex) is an abstract simplicial complex of dimension at most two: a finite set of 0-simplices called vertices, 1-simplices called edges, and 2-simplices called triangles. Each edge is a pair of vertices, and each triangle is a triple of vertices; moreover, each subset of size two in a triangle must be an edge.

Each 2-complex C corresponds naturally to a topological space, obtained as follows: Start with one point per vertex in C ; connect them by segments as indicated by the edges in C ; similarly, for every triangle in C , create a triangle whose boundary is made of the three edges contained in that triangle. By abuse of language, we identify C with that topological space.

Graph embeddings

Each graph has a natural associated topological space (for graphs without loops or multiple edges, this is a specialization of the definition for 2-complexes). An embedding Γ of a graph G into a 2-complex C is an injective continuous map from (the topological space associated to) G to (the topological space associated to) C . A face of Γ is a connected component of the complement of the image of Γ in C .

3

2-complexes and their data structures

Some preprocessing

A 3-book is a topological space obtained from three triangles by considering one side per triangle and identifying these three sides together into a single edge. We say that a 2-complex C contains a 3-book if C contains three distinct triangles that share a common edge. Because every graph can be embedded into a 3-book [6, Proposition 3.1], we have the following proposition:

Figure 1 On the left: A 2-complex with 5 singular points, numbered from 1 to 5, and 2 isolated edges (one between 3 and 4 and one between 1 and 2) where, at singular points, the cones are in green and the corners in yellow. On the right: the corresponding detached surface.

▶ Proposition 3.1. To decide the embeddability of a graph G on a 2-complex C , we can without loss of generality, after a linear-time preprocessing, assume the following properties on the input:

C has no 3-book and no connected component that is reduced to a single vertex; G has no connected component reduced to a single vertex, and at most one connected component homeomorphic to a segment. In the rest of this article, without loss of generality, we implicitly assume that C and G satisfy the properties stated in Proposition 3.1.

Structure of 2-complexes without 3-book or isolated vertex

Let C be a 2-complex without 3-book or isolated vertex, and let p be a vertex of C . Following [6, Section 2.2], we describe the possible neighborhoods of p in C . A cone at p is a cyclic sequence of triangles t 1 , . . . , t k , t 1 (k ≥ 3), all incident to p, such that, for each i = 1, . . . , k, the triangles t i and t i+1 (where t k+1 = t 1 ) share an edge incident with p, and any other pair of triangles have only p in common. A corner at p is a sequence of distinct triangles t 1 , . . . , t k , all incident to p, such that, for each i = 1, . . . , k -1, the triangles t i and t i+1 share an edge incident with p, any other pair of these triangles have only p in common, and no other triangle in C shares an edge incident with p and belonging to one of t 1 , . . . , t k . An isolated segment at p is an edge incident to p but not incident to any triangle. The cones, corners, and isolated segments at p form the link components at p.

The set of edges and triangles incident with a given vertex p of C are uniquely partitioned into cones, corners, and isolated segments. We say that p is a regular point if all the edges and triangles incident to p form a single cone or corner; in that case, p has an open neighborhood homeomorphic to a disk or a closed half-disk. Otherwise, p is a singular point. See Figure 1, left, for an illustration.

Detaching a singular point p in C consists of the following operation: replace p with new vertices, one for each cone, corner, and isolated segment at p. Detaching all singular points of a 2-complex (without 3-book) yields a disjoint union of (1) isolated segments and (2) a surface, possibly disconnected, possibly with boundary, called the detached surface (see Figure 1, right). The trace of the singular points on the detached surface are the marked points. Conversely, C can be obtained from a surface (possibly disconnected, possibly with boundary) and a finite set of segments by choosing finitely many subsets of points and identifying the points in each subset together.

The boundary of C is the closure of the set of points of C that have an open neighborhood homeomorphic to a closed half-plane. Equivalently, it is the union of the edges of C incident with a single triangle.

Topological data structure for 2-complexes

Any 2-complex C without 3-book or isolated vertex is obtained from the detached surface and a set of disjoints segments, by identifying finitely many finite subsets of points. It is thus easy to describe a topological data structure for storing 2-complexes, by storing the topology of the detached surface and the segments, and recording additionally the identification of points to obtain the complex. The size of C is the sum of the number of isolated segments, the number of connected components of the detached surface, the total genus of the detached surface, the total number of boundary components of the detached surface, and the total number of marked points of the detached surface (the occurrences of the singular points). This is, up to a constant factor, the size of the topological data structure indicated above, if the genus is stored in unary. Any 2-complex described in the usual combinatorial way can be converted in polynomial time into this representation. Thus, in the rest of this article, without loss of generality, we implicitly assume that C is given in the form of the above topological data structure.

Moreover, given two 2-complexes in the form above, deciding whether they are homeomorphic essentially amounts to testing whether the corresponding data structures are isomorphic, which leads to the following lemma (the running time might be improvable, but this suffices for our purposes):

▶

Proper and cellular graph embeddings on 2-complexes

Let C be a 2-complex with size c, G a graph, and Γ an embedding of G on C . The embedding Γ is proper if: the image of Γ meets the boundary of C only on singular points; the vertices of Γ cover the singular points of C .

The embedding Γ is cellular if each face of Γ is an open disk plus possibly some part of the boundary of C . We emphasize that this definition slightly departs from the standard one. Moreover, we will only consider cellular embeddings that are proper.

We will use a data structure to store possibly non-cellular embeddings of graphs on surfaces [5, Section 2.2]. Such a data structure is based on the gem representation of cellular graph embeddings [9, Section 2], but store additional information about the topology of the faces. It is important to remark that this data structure also allows to recover the topology of the underlying surface.

Let Γ be a proper graph embedding of a graph G on a 2-complex C (under the assumptions of Proposition 3.1). Let S be the detached surface of C . Because Γ is proper, it naturally induces an embedding Γ ′ , of another graph G ′ , on S ; some vertices of G located on singular points of C are duplicated in G ′ , the vertices of G located in the relative interior of isolated segments are absent from G ′ , and the edges of G not in G ′ are edges on the isolated segments of C . Our data structure, called combinatorial map, for storing the graph embedding Γ and the 2-complex C consists of storing (1) the graph embedding Γ ′ on S , as indicated in the previous paragraph, (2) the isolated segments of C , together with, for each such isolated segment, an ordered list alternating vertices and edges of Γ (or, instead of an edge, a mark indicating the absence of such an edge in the region of the isolated segment between the incident vertices), (3) the identifications of vertices of Γ ′ that are needed to recover Γ (and thus implicitly C ).

E S A 2 0 2 1

32:8 An FPT Algorithm for the Emebddability of Graphs into 2-Complexes

Isomorphisms between combinatorial maps are defined in the obvious way, similar to the concept of isomorphism between topological data structures: Two combinatorial maps are isomorphic if there is an isomorphism between the combinatorial maps restricted to the detached surfaces, isomorphisms between the maps on each isolated segments, and such that incidences are preserved on the singular points. We can easily test isomorphism between two combinatorial maps of size k and k ′ , respectively, in (k + k ′ ) O(k+k ′ ) time.

We will need an algorithm to enumerate all proper embeddings of small graphs on a given 2-complex. This is achieved by a brute-force algorithm: ▶ Lemma 3.3. Let C be a 2-complex of size c and k an integer. We can enumerate the (c + k) O(c+k) combinatorial maps of graphs with at most k vertices and at most k edges properly embedded on C in (c + k) O(c+k) time.

Partitioning graphs

Let C be a 2-complex and G a graph, which satisfy the properties of Proposition 3.1. In this section, we lay the structural foundations of the dynamic programming algorithm, described in the next section (Proposition 5.1). The goal, in this section and the following one, is to obtain an algorithm that takes as input C and G, and, in time FPT in the size of C and the branchwidth of G, reports correctly one of the following two statements: G has no proper cellular embedding on C , G has an embedding on C .

This algorithm uses dynamic programming on a rooted branch decomposition of G. When processing a node of the rooted branch decomposition, it considers embeddings of the subgraph of G induced by the edges in the leaves of the subtree rooted at that node in a region of C . This region will be delimited by a partitioning graph embedded on C . Our dynamic program will roughly guess the partitioning graph at each node of the rooted branch decomposition. For this purpose, we need that, if G has a proper cellular embedding on C , it has such an embedding that is sparse: at each node of the rooted branch decomposition of G, the partitioning graph corresponding to the embedding of the induced subgraph is small (its size is upper-bounded by a function of the branchwidth of G and of the size of C ). The goal of this section is to prove that this is indeed the case.

Let (E 1 , . . . , E k ) be an (ordered) partition of the edge set E(G) of G. (We will only use the cases k = 2 or k = 3.) The middle set of (E 1 , . . . , E k ) is the set of vertices of G whose incident edges belong to at least two sets E i .

Let Γ be a proper cellular embedding of G on C . Since Γ is cellular, every boundary of C is incident to at least one vertex of Γ. Let Γ be obtained from Γ by adding edges as follows: for any pair of vertices u and v of Γ consecutive along a given boundary component of C , we connect u and v via a new edge that runs along the boundary component. For each (ordered) partition (E 1 , . . . , E k ) of the edge set of G, we let Ê1 be the union of E 1 and of these new edges, and Êi = E i for each i ̸ = 1; thus, ( Ê1 , . . . , Êk ) is a partition of the set of edges of Γ.

The partitioning graph Π(Γ, E 1 , . . . , E k ) (or more concisely Π) associated to Γ and (E 1 , . . . , E k ) is a graph properly embedded on C (but possibly non-cellularly), with labels on its faces, defined as follows:

The vertex set of Π is the union of the singular points of C and of (the images under Γ of) the middle set of E 1 , . . . , E k . The relative interiors of the edges of Π are disjoint from the edges of Γ and from the isolated segments of C . Let f be a face of Γ (which is homeomorphic to an open disk plus possibly some points of the boundary of C ). Let us describe the edges of Π inside f . If, for some i ∈ {1, . . . , k}, the boundary of f is comprised only of edges of Γ that lie in a single set Êi , then Π contains no edge inside f . Otherwise, the boundary of f is a succession of edges of Ê1 , Ê2 , . . . , Êk . The edges of Π inside f run along the boundary of f ; for each i ∈ {1, . . . , k}, for each (maximal) group of consecutive edges in Êi along the boundary of f , we create an edge of Π that runs along this group, with endpoints the corresponding vertices on the boundary of f (see Figures 2 and3). These vertices either are in the middle set of (E 1 , . . . , E k ), or lie on the boundary of C (and thus on singular points of C ). It follows from the construction that Γ and Π intersect only at common vertices. Each face of Π is labelled by an integer in {0, . . . , k} as follows: faces of Π containing edges in Êi are labelled i, and the other ones are labelled 0. By construction of the graph Π, each face of Π contains edges from at most one set Êi , so this labelling is well defined.

Henceforth, we fix a rooted branch decomposition B of G, the root of which is denoted by ρ. Every arc α of B naturally partitions E(G) into two parts E 1 and E 2 , in which E 1 is the part on the same side as ρ; this (ordered) partition is the edge partition associated to α. Recall that Γ is a proper and cellular embedding of G on C ; we let Π(Γ, α) be Π(Γ, E 1 , E 2 ). Similarly, every node ν of B naturally partitions E(G) into three parts E 1 , E 2 , and E 3 , in which E 1 is the part on the same side as ρ; this partition is the edge partition associated to ν; we let Π(Γ, ν) be Π(Γ, E 1 , E 2 , E 3 ).

We say that Γ is sparse (with respect to B) if the following conditions hold, letting c be the size of C and w the width of B: Figure 4 Left: A vertex with 6 intervals, numbered from 1 to 6. Middle: The cyclic order obtained by applying the first type of simplification operation on intervals 1 and 2. After the simplification, the intervals 1 and 3 are merged into a single one, and similarly for the intervals 2 and 6. Right: The cyclic order obtained by applying the second type of simplification to the configuration on the left, on pairs of intervals {1, 2} and {4, 5}. After the simplification, the intervals 1, 3, and 5 are merged, and similarly for the intervals 2, 6, and 4.

For each arc α of B, the graph Π(Γ, α) has at most 74c + 26w edges; similarly, for each internal node ν of B, the graph Π(Γ, ν) has at most 3(74c + 26w) edges.

The result of this section is the following. ▶ Proposition 4.1. Let C be a 2-complex and G a graph, which satisfy the properties of Proposition 3.1. Let B be a rooted branch decomposition of G. Assume that G has a proper cellular embedding on C . Then it has a proper cellular embedding Γ on C that is sparse (with respect to B).

Monogons and bigons

A monogon of a graph Π embedded on a 2-complex C is a face of Π that is an open disk whose boundary is a single edge of Π (a loop). Similarly, a bigon of Π is a face of Π that is an open disk whose boundary is the concatenation of two edges of Π (possibly the same edge appearing twice). The following general lemma on graphs embedded on surfaces without monogons or bigons will be used.

▶ Lemma 4.2. Let S be a surface of genus g without boundary. Let Π be a graph embedded on S , not necessarily cellularly. Assume that Π has no monogon or bigon. Then |E(Π)| ≤ max{0, 3g + 3|V (Π)| -6}.

Vertex simplifications

The proof of Proposition 4.1 starts with any proper cellular embedding of Γ, and iteratively changes the cyclic ordering of edges around vertices in a specific way. Let (E 1 , E 2 ) be an (ordered) partition of E(G), let v be a vertex of G, and let C be a link component at v (if the image of v under Γ is a singular point, there may be several such link components). We restrict our attention to the edges of Γ incident to v and belonging to C, in cyclic order around v. For i = 1, 2, an interval (at v, relatively to ( Ê1 , Ê2 )) is a maximal contiguous subsequence of edges in this cyclic ordering that all belong to Êi ; the interval is labelled i. Simplifying v (with respect to (E 1 , E 2 )) means changing the cyclic ordering of the edges of Γ incident to v in C by one of the two following operations (Figure 4): 1. either exchanging two consecutive intervals in that ordering, in a way that the ordering of the edges in each interval is preserved; this operation is allowed only if v is incident to at least four intervals; 2. or performing the previous operation twice, on two disjoint pairs of consecutive intervals in that ordering; this allowed only if v is incident to at least six intervals. We will rely on the following lemma.

▶ Lemma 4.3. Let Γ be a proper cellular embedding of G on C , and let (E 1 , E 2 ) be an (ordered) partition of E(G). Let Γ ′ be another proper cellular embedding of G, obtained from Γ by simplifying one or two vertices with respect to (E 1 , E 2 ), while keeping the other cyclic orderings unchanged. Then:

1. |E(Π(Γ ′ , E 1 , E 2 ))| < |E(Π(Γ, E 1 , E 2 ))|; 2. for each (ordered) partition ( Ẽ1 , Ẽ2 ) of E(G) such that Êi ⊆ Êj for some i, j ∈ {1, 2}, we have |E(Π(Γ ′ , Ẽ1 , Ẽ2 ))| ≤ |E(Π(Γ, Ẽ1 , Ẽ2 ))|.
Proof. The proof is based on the following easy but key observations (the second one will be reused later): A simplification of v strictly decreases the number of intervals at v; the number of half-edges of Π(Γ, E 1 , E 2 ) at v in the link component C equals twice the number of intervals associated to ( Ê1 , Ê2 ) at v in C. The first point of the lemma immediately follows. For the second point, let us consider, in the cyclic ordering around v in C, a maximal contiguous sequence of edges in Êi . Since Êi ⊆ Êj , when simplifying with respect to (E 1 , E 2 ), this sequence is still contiguous in the new embedding Γ ′ . It follows that the number of intervals associated to ( Ê1 , Ê2 ) does not increase when replacing Γ with Γ ′ . ◀

Rearranging Γ with respect to an edge partition

We can now prove the following lemma:

▶ Lemma 4.4.
Let Γ be a proper cellular embedding of G on C , and let (E 1 , E 2 ) be an (ordered) partition of E(G). There exists a proper cellular embedding Γ ′ of G such that:

|E(Π(Γ ′ , E 1 , E 2 ))| ≤ 74c + 26w, where w is the size of the middle set of (E 1 , E 2 ); for each (ordered) partition ( Ẽ1 , Ẽ2 ) of E(G) such that Êi ⊆ Êj for some i, j ∈ {1, 2}, we have |E(Π(Γ ′ , Ẽ1 , Ẽ2 ))| ≤ |E(Π(Γ, Ẽ1 , Ẽ2 ))|.

Sketch of proof.

Let Π := Π(Γ, E 1 , E 2 ). We assume that Π has "many monogons or bigons" and show that there is another cellular embedding Γ ′ of G such that:

|E(Π(Γ ′ , E 1 , E 2 ))| < |E(Π(Γ, E 1 , E 2 ))|;
for each (ordered) partition ( Ẽ1 , Ẽ2 ) of E(G) such that Êi ⊆ Êj for some i, j ∈ {1, 2}, we have |E(Π(Γ ′ , Ẽ1 , Ẽ2 ))| ≤ |E(Π(Γ, Ẽ1 , Ẽ2 ))|. Intuitively, if Π has many monogons attached to a single vertex, or many bigons glued together, we can change the embedding Γ in a way that leads to vertex simplifications. By repeatedly iterating this argument, and up to replacing Γ with Γ ′ , this implies that we can assume without loss of generality that Π has "not too many monogons or bigons". Lemma 4.2 then implies that Π has at most 74c + 26w edges, which concludes. Immediately after applying the above procedure to an arc α of B, corresponding to the (ordered) partition ( Ẽ1 , Ẽ2 ) of E(G), we have |E(Π(Γ, Ẽ1 , Ẽ2 ))| ≤ 74c + 26w. We now prove that subsequent applications of Lemma 4.4 to other arcs of the rooted branch decomposition do not increase this number of edges. Indeed, let α be another arc, corresponding to the (ordered) partition (E 1 , E 2 ) of E(G), to which we apply Lemma 4.4. The arc α partitions the nodes of the tree B into two sets N 1 and N 2 , and similarly α partitions the nodes of the tree B into two sets Ñ1 and Ñ2 . Because B is a tree, we have Ñi ⊆ N j for some i, j ∈ {1, 2}. This implies that Êi ⊆ Êj for some i, j ∈ {1, 2}; so the second item of Lemma 4.4 implies that the number of edges of Π(Γ, Ẽ1 , Ẽ2 ) does not increase when processing arc α. This proves the claim.

Finally, there remains to prove that, for each internal node ν of B, the graph Π(Γ, ν) has at most 3(74c + 26w) edges. This follows relatively easily from the above claim. ◀

Dynamic programming algorithm

The result of this section is the following proposition. (Proposition 4.1 implies that we can remove the adjective "sparse" in the above proposition.)

Bounding graphs

Let B be a rooted branch decomposition of G of width w. Recall (see Section 2.1) that the root ρ of B is a leaf associated to no edge of G. Our algorithm will use dynamic programming in the rooted branch decomposition. For each arc α of B, let G α be the subgraph of G induced by the edges of G corresponding to the leaves of the subtree of B rooted at α. The general idea is that we compute all possible relevant embeddings of G α in subregions of C . Such subregions will be delimited by a graph embedded on C of small complexity. For the dynamic program to work, we also need to keep track of the location of the vertices in the middle set of α. More precisely, a bounding graph for G α is a proper labelled graph embedding Π on C (but possibly non-cellular), such that: some vertices of Π are labelled; these labels are exactly the vertices of the middle set associated with α, and each label appears exactly once; each unlabelled vertex of Π is mapped to a singular point of C ; each face of Π is labelled 0, 1, or 2;

32:13

G α has an embedding Γ α that respects Π: each vertex of Π labelled v is mapped, under Π, to the image of v in Γ α ; moreover, the relative interior of each edge of Γ α lies in the interior of a face of Π labelled 2.

A bounding graph for G α is sparse if it has at most 74c + 26w edges. Remark that, if Γ is a sparse proper cellular embedding of G on C (as defined in Section 4), then Π(Γ, α) is a sparse bounding graph for the restriction of Γ to G α .

Henceforth, we regard two (labelled) properly embedded graphs as equal if and only if their (labelled) combinatorial maps are isomorphic. A list L α of sparse bounding graphs for G α is exhaustive if the following condition holds: If G has a sparse proper cellular embedding on C , then for each such embedding Γ, the (combinatorial map of the) graph Π(Γ, α) is in L α .

The induction step for the dynamic programming algorithm is the following.

▶ Proposition 5.2. Let ν be a non-root node of B and α be the arc of B incident to ν that is the closest to the root ρ. Assume that, for each arc β ̸ = α of B incident to ν, we are given an exhaustive list of sparse bounding graphs for G β . Then we can, in (c + w) O(c+w) time, compute an exhaustive list of (c + w) O(c+w) sparse bounding graphs for G α .

Assuming Proposition 5.2, the proof of which is deferred to the next subsection, it is easy to prove Proposition 5.1:

Proof of Proposition 5.1, assuming Proposition 5.2. We apply the algorithm of Proposition 5.2 in a bottom-up manner in the rooted branch decomposition B. Let α be the arc of B incident with the root node ρ. We end up with an exhaustive list of sparse bounding graphs for G α = G. By definition of a bounding graph, if this list is non-empty, then G has an embedding on C . On the other hand, by definition of an exhaustive list, if this list is empty, then G has no sparse proper cellular embedding on C .

There are O(n) recursive calls, each of which takes (c + w) O(c+w) time. ◀

The induction step: Proof of Proposition 5.2

Proof of Proposition 5.2. First case. Let us first assume that ν is a (non-root) leaf of B; thus, G α is a single edge uv. Using Lemma 3.3, we compute all the labelled combinatorial maps of sparse bounding graphs for G α . This is clearly an exhaustive list. Indeed, assume that G has a sparse proper cellular embedding Γ on C ; by sparsity, Π(Γ, α) has at most 74c + 26w edges; thus, one of the labelled combinatorial maps computed will be equal to that of Π(Γ, α).

Second case. Let us now assume that ν is an internal node of B. As above, let α be the arc of B incident to ν that is the closest to the root ρ. Let β and γ be the arcs different from α incident to ν. Let L β and L γ be exhaustive lists of bounding graphs for G β and G γ , respectively. Intuitively, every pair of bounding graphs in L β and L γ that are compatible, in the sense that the regions labelled 2 in each of these two graphs are disjoint, will lead to a bounding graph in L α . This is the motivating idea to our approach. More precisely, we will enumerate labelled combinatorial maps Π, each of which can be "restricted" to two compatible graphs, which are possible bounding graphs for G β and G γ . If these two restrictions lie in L α and L β , this leads to a graph that is added to L α .

We first introduce some terminology. Let Π be a graph properly embedded on C (possibly non-cellularly), with faces labelled 0, 1, 2, or 3, and with labels on some vertices. Let i, j, k be integers such that {i, j, k} = {1, 2, 3}. We will define a graph embedding Π i,j obtained E S A 2 0 2 1 32:14 An FPT Algorithm for the Emebddability of Graphs into 2-Complexes from Π by somehow "merging" faces i and j. First, for an illustration, refer back to Figures 2 and3: If Π is the graph embedding depicted on the right of Figure 3, then the configurations shown on the right of Figure 2 correspond, from top to bottom, to Π 2,3 , Π 1,3 , and (Π 1,2 ) - (the latter being the graph Π 1,2 in which each face label 3 is replaced by a 2).

Formally, Π i,j is defined as follows. First, let us replace all face labels j by i. Now, for each face f of Π that is homeomorphic to a disk and labelled 0, we do the following. The boundary of f is made of edges of Π; for the sake of the discussion, let us temporarily label each such edge by the label of the face on the other side of f . If all edges on the boundary of f all labelled i, then we remove all these edges, and f becomes part of a larger face labelled i. Otherwise, for each maximal subsequence e 1 , . . . , e ℓ of edges along the boundary of f that are all labelled i, we remove each of e 1 , . . . , e ℓ , and replace them with an edge inside f from the source of e 1 to the target of e ℓ . Finally, we remove all isolated vertices that do not coincide with a singular point of C , and all vertices in the relative interior of an isolated segment that are incident to two faces with the same label.

For any labelled combinatorial map Π, we denote by Π -the same map where each label 3 on a face is replaced by a 2. The easy but key properties of this construction are the following:

(i) Assume that Π 1,3 is a bounding graph for G β and (Π 1,2 ) -is a bounding graph for G γ .

Then Π 2,3 is a bounding graph for G α .

(ii) The node ν naturally partitions the edge set of G into three parts, which we denote by E 1 (on the side of α), E 2 (on the side of β), and E 3 (on the side of γ). Assume that G has a sparse proper cellular embedding Γ on C and that Π = Π(Γ, E 1 , E 2 , E 3 ). Then: We compute our exhaustive list L α of sparse bounding graphs for G α as follows. Initially, let this list be empty. Using Lemma 3.3, we enumerate all combinatorial maps Π of graphs with at most c + 3w vertices and 3(74c + 26w) edges properly embedded on C (possibly non-cellularly), with faces labelled 0, 1, 2, or 3, and such that the labels appearing on the vertices are exactly the vertices of the middle set of α, β, or γ (and each label appears exactly once). This takes (c + w) O(c+w) time. Whenever Π 1,3 ∈ L β and (Π 1,2 ) -∈ L γ , we add Π 2,3 to L α . Finally, we eliminate duplicates by testing pairwise isomorphism between the labelled combinatorial maps in L α , and remove the graphs that are not sparse or contain vertices that bear a label not in the middle set of α.

Π(Γ, α) = Π(Γ, E 1 , E 2 ∪ E 3 ) = Π
L α contains only sparse bounding graphs for G α , by (i) above. Moreover, let Γ be a sparse proper cellular graph embedding of G on C . By sparsity, one of the graphs Π enumerated in the previous paragraph is Π(Γ, ν). By definition of L β and L γ , we have that Π(Γ, β) ∈ L β and Π(Γ, γ) ∈ L γ , so by (ii) above, Π(Γ, α) ∈ L α , which implies that L α is exhaustive. ◀

Lemma 3 . 2 .

 32 Given two 2-complexes C and C ′ , given in the topological representation above, of sizes c and c ′ respectively, we can decide whether C and C ′ are homeomorphic in time (c + c ′ ) O(c+c ′ ) .

Figure 2

 2 Figure 2 Construction of the partitioning graph Π = Π(Γ, E1, E2), for three choices of the partition (E1, E2) of the same embedding Γ. Only a part of the 2-complex C is shown, with a boundary at the upper part, and without singular point. Left: The graph embeddings Γ (in thick lines) and Π (in thin lines). Right: The sole graph Π, together with the labelling of its faces.

Figure 3

 3 Figure 3 The partitioning graph Π = Π(Γ, E1, E2, E3). Left: The graph embeddings Γ (in thick lines) and Π (in thin lines). Right: The sole graph Π, together with the labelling of its faces.

1 Proof of Proposition 4 . 1 .

 141 Let B be a rooted branch decomposition of G, and let Γ be a proper cellular embedding of G on C . We consider each arc α of the rooted branch decomposition in turn, in an arbitrary order. For each such arc, we modify Γ by applying Lemma 4.4. We first claim that after these iterations, for each arc α of B, we have |E(Π(Γ, α))| ≤ 74c + 26w.

▶ Proposition 5 . 1 .

 51 Let C be a 2-complex and G a graph, which satisfy the properties of Proposition 3.1. Let c be the size of C and n the number of vertices and edges of G. Let B be a rooted branch decomposition of G of width w. In (c + w) O(c+w) n time, one can report one of the following statements, which is true: G has no sparse proper cellular embedding into C ; G has an embedding into C .

2 , 3 ;

 23 Π(Γ, β) = Π(Γ, E 1 ∪ E 3 , E 2 ) = Π 1,3 ; Π(Γ, γ) = Π(Γ, E 1 ∪ E 2 , E 3 ) = (Π 1,2 ) -.Property (ii) is, again, illustrated by Figures2 and 3: If (E 1 , E 2 , E 3 ) is the edge partition depicted on Figure3, then the edge partitions depicted on Figure2, left, are, respectively, (E 1 , E 2 ∪E 3 ), (E 1 ∪E 3 , E 2 ), and (E 1 ∪E 2 , E 3 ). As shown above, the corresponding partitioning graphs are respectively Π 2,3 , Π 1,3 , and Π - 1,2 . If Π is the graph embedding depicted on the right of Figure3, then the configurations shown on the right of Figure2correspond, from top to bottom, to Π 2,3 , Π 1,3 , and (Π 1,2 ) - (the latter being the graph Π 1,2 in which each face label 2 is replaced by a 2).

  for suitable introductions under various viewpoints. We recall some basic definitions and properties. A surface (without boundary) S is a compact, connected Hausdorff topological space in which every point has an open neighborhood homeomorphic to the open disk. Up to homeomorphism, every surface S is obtained from a sphere by: either removing g/2 open disks and attaching a handle (a torus with an open disk removed) to each resulting boundary component, where g is an even, nonnegative integer called the (Euler) genus of S ; in this case, S is orientable; or removing g open disks and attaching a Möbius band to each resulting boundary component, for a positive number g called the (non-orientable) genus of S ; in this case, S is non-orientable. A possibly disconnected surface is a disjoint union of surfaces. A surface with boundary is obtained from a surface (without boundary) by removing a finite set of interiors of disjoint closed disks. The boundary of each disk forms a boundary component of S . The genus of S is defined as the genus of the original surface without boundary. Equivalently, a surface with boundary is a compact, connected Hausdorff topological space in which every point has an open neighborhood homeomorphic to the open disk or the

Acknowledgements

We would like to thank Arnaud de Mesmay for useful discussions. 1

Funding Partially supported by the ANR projects Min-Max (ANR-19-CE40-0014) and SoS (ANR-17-CE40-0033).

https://arxiv.org/abs/2107.06236

A wall of size k × k is a subgraph of the (k × k)-grid obtained by removing alternately the vertical edges of even (resp. odd) x-coordinate in each even (resp. odd) line, and then the degree-one vertices.

We first use any algorithm to approximate the treewidth of G, e.g., Fomin et al. [11, Theorem 1.1]: In polynomial time, we either compute a branch decomposition of small width of G, in which case we are done, or correctly report that the treewidth is large. In the latter case, the result by Chekuri and Chuzhoy [START_REF] Chekuri | Polynomial bounds for the grid-minor theorem[END_REF] implies that there is a large grid minor, which we can compute in randomized polynomial time using the same article, or in deterministic f (k) • n 2 time, where f is some computable function and k is the size of the grid [START_REF] Robertson | Graph minors. XIII. The disjoint paths problem[END_REF]Algorithm 4.4]. We have thus computed a subdivision of a large wall.

We then partition this wall into Ω(c) smaller subwalls W 1 , . . . , W m , where m = Ω(c), each bounded by a cycle γ i . We then show that we can assume that there are (intuitively) not too many connections, in G, between two different walls W i , W j that avoid the cycles γ i E S A 2 0 2 1 32:16 An FPT Algorithm for the Emebddability of Graphs into 2-Complexes and γ j ; otherwise, by computing these connections, we exhibit a subgraph of G of genus Ω(c), and deduce that G is not embeddable on C . Finally, we compute a cycle γ of G, such that one connected component of G -γ is planar and contains a subdivision of a smaller, but still large, wall.

We then show, borrowing some ingredients to Kociumaka and Pilipczuk [15, Section 5], that the central vertex of this wall is irrelevant, in the sense that its removal does not affect the embeddability or non-embeddability of the graph into C . Intuitively, v is surrounded by Ω(c) concentric cycles in the wall; if G -v is embedded in C , then two concentric cycles must bound an annulus. The planar part inside the inner cycle can be embedded close to the boundary of the annulus that corresponds to this inner cycle.

Iterating the whole procedure, we obtain a graph of branchwidth polynomial in the size of C . ◀


Pascale Domingo

Luc Vervisch
email: luc.vervisch@insa-rouen.fr

Revisiting the relation between premixed flame brush thickness and turbulent burning velocities from Ken Bray's notes

Keywords: Turbulent premixed flames, Turbulent burning velocity, Turbulent flame brush, Bi-Modal-Limit theory

Applying the Bray-Moss-Libby (BML) bi-modal asymptotic limit to a progress variable signal in a premixed turbulent flame, a relation is established between the variation rate of the mean turbulent flame brush thickness, the flame surface wrinkling and the turbulent burning velocity. This scaling suggests that the link between the amplification of the flame surface by turbulence and the increase of the turbulent burning velocity may need to be modulated by a corrective term proportional to the rate of variation of the turbulent flame brush thickness. This correction is expected to be driven by a short relaxation time scale, which is representative of the turbulent flame brush adjusting to its continuously varying turbulence environment. The analysis is grounded on the usual exact expressions for the various characteristic displacements and consumption speeds, widely used in turbulent combustion modeling. Prior to this analysis, the validity of the implication on these characteristic speeds of the thin flamelet framework, introduced to perform the derivation of the relation between the turbulent flame brush dynamics and the turbulent burning velocity, is evaluated from direct numerical simulation (DNS) of a turbulent premixed jet-flame.

Introduction

Back twenty years ago, Prof. Ken Bray was used to visit Normandy every year and we were lucky to enjoy endless discussions on flame physics and turbulent combustion modeling.

The turbulent burning velocity was a recurrent subject, leading to notes that were left unpublished. Mainly because some of these results were quite intriguing, unfortunately without directly leading to novel modeling ideas ready for simulating flames, and all of us were, at that time, more attracted by the rapidly growing Large Eddy Simulation (LES) tools than by dealing with additional manipulations of those 'old fashion concepts'. However, in turbulent reacting flow physics there is no getting away from certain facts and the vast literature on the subject (see [START_REF] Bradley | Flame stretch rate as a determinant of turbulent burning velocity[END_REF][START_REF] Peters | Turbulent Combustion[END_REF][START_REF] Borghi | Turbulent premixed combustion: Further discussions on the scales of fluctuations[END_REF][START_REF] Corvellec | Turbulent premixed flames in the flamelet regime: burning velocity spectral properties in the presence of countergradient diffusion[END_REF][START_REF] Charlette | A power-law flame wrinkling model for LES of premixed turbulent combustion Part I: non-dynamic formulation[END_REF][START_REF] Charlette | A power-law flame wrinkling model for LES of premixed turbulent combustion Part II: dynamic formulation[END_REF][START_REF] Bradley | Correlation of turbulent burning velocities of ethanol-air, measured in a fan-stirred bomb up to 1.2MPa[END_REF][START_REF] Hawkes | A petascale direct numerical simulation study of the modelling of flame wrinkling for large-eddy simulations in intense turbulence[END_REF][START_REF] Haq | Wrinkling and curvature of laminar and turbulent premixed flames[END_REF][START_REF] Colin | A thickened flame model for Large Eddy Simulations of turbulent premixed combustion[END_REF][START_REF] Tamadonfar | Flame brush characteristics and burning velocities of premixed turbulent methane/air Bunsen flames[END_REF][START_REF] Verma | Does sensitivity of measured scaling exponents for turbulent burning velocity to flame configuration prove lack of generality of notion of turbulent burning velocity?[END_REF][START_REF] Bray | The interaction between turbulence and chemistry in premixed turbulent flames[END_REF][START_REF] Bray | Finite rate chemistry and presumed PDF models for premixed turbulent combustion[END_REF][START_REF] Cheng | On defining the turbulent burning velocity in premixed V-shaped turbulent flames[END_REF][START_REF] Bray | Pre-mixed flames in stagnating turbulence: Part Vevaluation of models for the chemical source termm[END_REF][START_REF] Knudsen | A dynamic model for the turbulent burning velocity for large eddy simulation of premixed combustion[END_REF][START_REF] Attili | Turbulent flame speed and reaction layer thicknening in premixed jet flames at constant Karlovitz and increasing Reynolds numbers[END_REF][START_REF] Filatyev | Measured properties of turbulent premixed flames for model assessment, including burning velocities, stretch rates, and surface densities[END_REF][START_REF] Jiang | High-temperature, high-pressure burning velocities of expanding turbulent premixed flames and their comparison with Bunsen-type flames[END_REF] and multiple references therein), suggests that the turbulent burning velocity is one of them. How it relates to global turbulence and flame properties still motivates combustion scientists, bringing new ideas and novel challenges for real combustion systems. Thereby observations resulting from simple manipulations of the balance equation for a progress variable may still be of interest.

One of the things appearing when generalising to a turbulent flame brush the basic concepts of displacement and consumption speeds, well defined and widely used to address laminar premixed flames dynamics [START_REF] Peters | Turbulent Combustion[END_REF][START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF][START_REF] Law | Combustion Physics[END_REF][START_REF] Clavin | Combustion Waves and Fronts in Flows[END_REF], is the striking fact that the flame wrinkling factor (or the ratio between turbulent and laminar flame surfaces) should not be considered, and therefore modeled, independently to the scalar transport by velocity fluctuations. A constraint which may not be that easy to meet in Reynolds Average Navier Stokes (RANS) or LES of turbulent premixed flames.

Along similar lines, a relation between the turbulent burning velocity and the rate of growth of the flame brush thickness may be established. This relation is also extended to the turbulent displacement speed inside the turbulent flame brush. To the authors' knowledge, the link between these quantities has not been discussed so far under this form and it can help understanding the departure between the flame surface amplification by turbulence and the increase of the turbulent burning velocity sometimes reported, see for instance recent findings in [START_REF] Attili | Turbulent flame speed and reaction layer thicknening in premixed jet flames at constant Karlovitz and increasing Reynolds numbers[END_REF]. This paper is organised as follows, fundamentals on displacement and consumption speeds are first summarised for laminar and turbulent flows, using both Reynolds and density weighted (Favre) averaging. Then, these quantities are extracted from a turbulent premixed jet flame DNS database. After that, the thin flamelet approximation is introduced and it is examined to which extend its implication on the characteristics speeds is verified in the DNS. Finally, relations between the turbulent burning velocity, the laminar burning velocity, the flame wrinkling factor and the rate of change of the flame brush thickness are derived and discussed.

Displacement and consumption speeds

Laminar flow

Let us consider c(x, t) a progress variable, c = 0 in fresh gases and c = 1 in fully burnt products, defined under the assumption of unity Schmidt and Lewis numbers [START_REF] Bray | The Challenge of turbulent combustion[END_REF]. The flow velocity vector is u and V r (c) is the iso-c displacement speed, namely the propagation speed relative to the flow in the normal direction n = -∇c/|∇c| to the iso-c surface, thus the added velocity which allows c(x, t) to remain constant:

∂c ∂t + (u + V r n) • ∇c = 0 , (1)
so that u abs = u + V r n is the absolute velocity of the iso-c surface in the laboratory frame.

For a mass diffusive flux expressed as -ρD∇c where ρ is the density following the continuity equation (∂ρ/∂t) + ∇ • (ρu) = 0, D a diffusive coefficient, and a burning rate ωc , the c-budget may also be written

∂ρc ∂t + ∇ • (ρuc) = ∇ • (ρD∇c) + ωc = -ρV r n • ∇c = ρV r |∇c| , (2)
and

V r = ∇ • (ρD∇c) + ωc ρ|∇c| . (3)
The consumption speed of the flame S c is defined by an integration across the reaction zone along a direction normal to the iso-c surfaces

ρ o S c = +∞ -∞ ωc dn . (4)
In a steady, planar and unstrained one-dimensional premixed flame, mass conservation brings the relation between S o L , the laminar flame speed, the consumption speed and the relative progression velocity at the fresh gases leading edge S o L = S c = V r (c = 0) and also the relation with the displacement speed for all values of the progress variable

ρ o S c = ρ o S o L = ρ(c)V r (c) . (5)

Turbulent flow

With • denoting RANS averaging or LES filtering and • = ρ • /ρ the density weighted (Favre) counterpart, the following is obtained from above relations for either Reynolds av-eraging

∂c ∂t + u • ∇c + T c = 1 ρ (∇ • (ρD∇c) + ωc ) = V r |∇c| , (6)
with

T c = u • ∇c -u • ∇c, or Favre averaging ∂ρc ∂t + ∇ • (ρũc) + ∇ • τ c = ∇ • (ρD∇c) + ωc = ρV r |∇c| , (7)
where τ c = ρuc-ρũc. Then, introducing V T and V T the mean or filtered relative progression velocities of the iso-c and iso-c surfaces, respectively,

∂c ∂t + (u + V T n) • ∇c = 0 , (8)
∂c ∂t + (ũ + V T ñ) • ∇c = 0 , (9)
with n = -∇n/|∇n| and ñ = -∇ñ/|∇ñ|. The definition of the displacement speeds through the turbulent flame brush thus read

V T = 1 ρ (∇ • (ρD∇c) + ωc ) -T c |∇c| = V r |∇c| -T c |∇c| , (10)
V T = ∇ • (ρD∇c) + ωc -∇ • τ c ρ|∇c| = ρV r |∇c| -∇ • τ c ρ|∇c| . (11)
According to ( 8) and ( 9), the knowledge of V T (c) or of V T (c) at all the points within the flame brush, would provide a complete closure of the balance equation for the progress variables c or c. This constitutes the basis of widely used modeling for turbulent premixed flames in RANS context [START_REF] Zimont | Turbulent Flame Closure Model Combustion Model[END_REF]. A similar conclusion is reached introducing the so-called G-Equation [START_REF] Kerstein | Field equation for interface propagation in an unsteady homogeneous flow field[END_REF], which provides a unique framework to explore the dynamics of a turbulent flame brush, also in the LES context, see N. Peters textbook where numerous relations and modeling to simulate turbulent flames are examined along these lines [START_REF] Peters | Turbulent Combustion[END_REF].

Within the Reynolds averaging context, introducing the surface averaging [27, 28]

• s = • |∇c| |∇c| , (12)
leads to

V r |∇c| = V r s |∇c| , (13)
and with u = uu

T c = u • ∇c -u • ∇c = -(u • n|∇c| -u • n|∇c|) = -( u • n s -u • n s ) |∇c| , = -u • n s |∇c| . ( 14
)
Combining these relations with (10), V T may be expressed in terms of the unresolved flame surface wrinkling Ξ = |∇c|/|∇c|

V T = [ V r s + u • n s ] Ξ . (15)
Where, as expected, it is seen that the surface displacement speeds of the iso-c surfaces evolve with the flame surface wrinkling. In addition, u • n s , the transport by velocity fluctuations can increase or decrease V T relative to the contribution of V r s , the conditional surface mean of the displacement speed.

Direct numerical simulation database

A previously developed direct numerical simulation database (DNS) of a turbulent premixed jet flame is now probed to examine the above relations for the displacement speeds and their extension to the thin flame regime. The configuration of this DNS is from the experiment reported in [START_REF] Chen | The detailed flame structure of highly stretched turbulent premixed methane-air flames[END_REF][START_REF] Mansour | Highly strained turbulent rich methane flames stabilized by hot combustion products[END_REF], with a coflow temperature imposed at the adiabatic burnt gases temperature.

This DNS database was explored in [START_REF] Cifuentes | Local volumetric dilatation rate and scalar geometries in a premixed methane-air turbulent jet flame[END_REF] to study the local volumetric dilatation rate and progress variable field characteristic geometries and also in [START_REF] Cifuentes | Effects of the local flow topologies upon the structure of a premixed methane-air turbulent jet flame[END_REF] to investigate the effects of the local flow topologies upon the structure of the premixed turbulent flame. The same database served to investigate one-dimensional filtered flame sub-grid scale modeling in the light of an approximate deconvolution operation [START_REF] Domingo | DNS and approximate deconvolution as a tool to analyse one-dimensional filtered flame sub-grid scale modeling[END_REF]. In this former work, it was discussed how the strongly convoluted three-dimensional turbulent flame surface leads to significant departure from the one-dimensional flamelet response. This DNS provided also in [START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated manifolds using convolutional neural networks[END_REF] the basis to discuss direct mapping from LES resolved scales to filtered-flame generated manifolds using convolutional neural networks (CNN).

The detail on the generation of this DNS and its main features may be found in [START_REF] Domingo | DNS and approximate deconvolution as a tool to analyse one-dimensional filtered flame sub-grid scale modeling[END_REF] and are not repeated here for brevity, the main properties are summarised in Fig 1 . This simulation relies on the idea that a well-resolved LES of the close field of a turbulent premixed and piloted jet-flame generates a flow which can be used as the inlet plane of a DNS of the interaction between velocity fluctuations and a premixed reaction zone. The LES and the Premixed flame detailed tabulated chemistry is built with unity Schmidt numbers from the GRI-3.0 [35] methane-air mechanism with the progress variable defined form major products (CO, CO 2 , H 2 O) and NOx [START_REF] Godel | Tabulation of NOx chemistry for Large-Eddy Simulation of non-premixed turbulent flames[END_REF], in order to better reproduce the relaxation towards equilibrium fully burnt products featuring large chemical time scales. The LES part is simulated with a progress variable presumed probability density function (pdf) approach [START_REF] Domingo | Large-Eddy Simulation of a lifted methane-air jet flame in a vitiated coflow[END_REF] and the sub-grid scale (SGS) momentum fluxes are approximated with the Vreman model [START_REF] Vreman | An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications[END_REF].

The flow solver SiTCom [START_REF] Bouheraoua | Large-eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF] was used which considers the aerothermochemistry equations in their fully compressible form together with the balance equation for the filtered progress variable. The convective terms are discretised with a fourth-order centered skew-symmetric-like scheme [START_REF] Ducros | High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows[END_REF] and the diffusive terms with a fourth-order centered scheme. Time is advanced explicitly with a third order Runge-Kutta method and acoustic boundary conditions [START_REF] Lodato | Three-dimensional boundary conditions for Direct and Large-Eddy Simulation of compressible viscous flows[END_REF] are imposed at inlet and outlet, with the measured profiles with synthetic turbulence [START_REF] Klein | A Digital Filter Based Generation of Inflow Data for Spatially Developing Direct Numerical or Large Eddy Simulations[END_REF] prescribed at inlet.

Various quantities are extracted from this DNS database, with or without formulating the thin-flamelet approximation introduced in the next section. In a preliminary analysis, the distribution of |∇c|, the magnitude of the gradient of the progress variable, is given in Fig. 2. The scatter plot within the DNS zone of |∇c| versus the progress variable shows quite large variations within the flame. However, over the reaction zone (i.e., 0.8 < c < 0.9), the conditional mean |∇c| | c * (squares in Fig. 2) becomes close to G(c) (solid line), the gradient as observed in the one-dimensional laminar flame which served for tabulating the chemistry. Because of the specific progress variable used which was originally designed to also capture also slow NOx chemistry [START_REF] Godel | Tabulation of NOx chemistry for Large-Eddy Simulation of non-premixed turbulent flames[END_REF], c is slowly evolving towards unity in the burnt gases and G(c) almost vanishes well before c → 1. In the fresh gases and in the preheating zone, the turbulence impacts on the flame structure and |∇c| | c * becomes smaller than

G(c).

The histogram of the displacement speed ρV r computed from Eq. ( 3) and normalised by ρ o S L is given in Fig. 3. Even though a non-flamelet behavior was seen in the gradient plots, the statistics of V r stays well centred around the flamelet response (Eq. ( 5)). This is understood from Fig. 4, where snapshots of the various terms contributing to V r are displayed. Comparing Fig. 4(a) with Fig. 4(d) shows that the departure of |∇c| from the one-dimensional flamelet gradient G(c), occurs in zones where the flame surface is highly curved, with a pronounced thickening of the progress variable profiles.

This database built in the configuration of a real jet-flame thus combines typical flamelet and non-flamelet responses, which is expected for the Karlovitz numbers considered (Fig. 1).

Thin flamelet approximation and BML analysis

To introduce the turbulent burning velocity, which relates to an averaged mass burning rate, it is necessary to incorporate the density in the formulation and therefore to focus on Favre averaging or filtering. Because the flame wrinkling Ξ = |∇c|/|∇c| is defined from

Reynolds averaging, an explicity relation between c and c must also be used, such relation is readily available introducing the Bray Moss Libby (BML) theory in the analysis [START_REF] Bray | The Challenge of turbulent combustion[END_REF][START_REF] Bray | Unified modeling approach for premixed turbulent combustion -Part I: General formulation[END_REF].

Thin flamelet approximation

Introducing the thin flamelet approximation (Eq. ( 5)) into the Favre averaging expression (10) leads to

ρ V T = ρ o S o L |∇c| |∇c| - ∇ • τ c |∇c| . ( 16
)
Figures 5 show ρ V T computed from Eq. 11 in the jet-flame DNS. The averaging procedure is done over vertical lines (streamwise direction) as in [START_REF] Attili | Turbulent flame speed and reaction layer thicknening in premixed jet flames at constant Karlovitz and increasing Reynolds numbers[END_REF], here to get 343×343 distributions of ρ V T and c from averaging over the 243 points of the vertical direction. Then ρ V T |c , the conditional mean versus the density weighted progress variable, is calculated over the 117649 averaged values to visualise the mean response through the flame brush. The circles in these plots are the values obtained from the thin flamelet approximation (RHS of ( 16)).

A good agreement is found between ρ ṼT from [START_REF] Tamadonfar | Flame brush characteristics and burning velocities of premixed turbulent methane/air Bunsen flames[END_REF] and its thin flame expression [START_REF] Bray | Pre-mixed flames in stagnating turbulence: Part Vevaluation of models for the chemical source termm[END_REF]. The contribution of ∇ • τ c , the divergence of the transport by unresolved velocity fluctuation (dashed line in Fig. 5), far from being negligible and approximating ρ V T without it would lead to error in the estimation of the relative progression velocity of the iso-c. Selecting

Bi-Modal-Limit analysis

Within the context of the strict bi-modal-limit (BML) theory [START_REF] Bray | The Challenge of turbulent combustion[END_REF][START_REF] Bray | Unified modeling approach for premixed turbulent combustion -Part I: General formulation[END_REF], the probability density function (PDF) of the progress variable reads

P (c * ) = αδ(c * ) + βδ(1 -c * ) . ( 17
)
PDF normalisation implies α + β = 1 and

c = 1 0 c * P (c * )dc * = β , ( 18
)
ρc = ρc = 1 0 ρ(c * )c * P (c * )dc * = ρ b β , (19)
therefore β = c = ρc/ρ b . Then, ρ = 1 0 ρ(c * )P (c * )dc * = 1 - ρc ρ b ρ o + ρc ρ b ρ b = ρ o + ρc(1 -ρ o /ρ b ) = ρ o -ρτ c , ( 20
)
with τ = (ρ o /ρ b ) -1 = (T b /T o ) -1. From (20) ρ = ρ o /(1 + τ c
) and c = ρc/ρ b , the relation between c and c is obtained:

c = (1 + τ )c 1 + τ c , (21)
also,

|∇c| = ρ o ρ 2 1 1 + τ |∇c| , (22)
and Eq. ( 16) becomes

ρ V T = (1 + τ ) ρ ρ o 2 ρ o S o L Ξ - ∇ • τ c |∇c| . ( 23
)
In the particular case of a steady and planar turbulent flame brush, from mass conservation the turbulent burning velocity S T is defined so that

ρ V T = ρ o S T . ( 24
)
Therefore the product ρ(c) V T (c) should not depend on c and from the relation ( 23), the flame wrinkling, Ξ, and the transport by unresolved velocity fluctuations ∇ • τ c should not, in theory, be modelled independently of each other to secure this weak dependence on c.

The fact that the product ρ(c) V T (c) should not depend much on c cannot strickly be verified in the post-processing of the DNS considered in this study, since it includes finiterate chemistry and therefore cannot exhibit the asymptotic bi-modal behavior. However, Figs. 5 and6 show that ρ(c) V T (c) weakly varies over a quite large range of variation of c. Therefore, it appears that the implications of the bi-modal analysis on the statistical behavior of the flame are in fact more robust than expected. Accordingly, it makes sense to adopt the above BML formalism to pursue the analysis towards the dynamics of the flame brush thickness.

Flame surface in the Bi-Modal-Limit

The surface density function Σ(c * ) = |∇c||c * P (c * ) is introduced [START_REF] Pope | The evolution of surfaces in turbulence[END_REF][START_REF] Pope | Computations of turbulent combustion: progress and challenges[END_REF][START_REF] Vervisch | Surface density function in premixed turbulent combustion modeling, similarities between probability density function and flame surface approaches[END_REF] so that

|∇c| = Ξ|∇c| = 1 0 Σ(c * )dc * = 1 0 |∇c||c * P (c * )dc * , (25)
where • |c * denotes a conditional average. The turbulent consumption speed Sc may then be written

ρ o Sc = +∞ -∞ ωc dξ = 1 0   1 0 ωc P (c * )dc *   dc |∇c| , (26)
= 1 0   1 0 ωc Σ(c * ) |∇c||c * dc *   dc |∇c| , (27)
where ξ denotes the coordinate in the direction normal to the reaction zone, assuming that the iso-scalar surfaces are quasi-parallel. In the thin reaction zone limit, Σ(c * ) ≈ |∇c| and

|∇c||c * ≈ dc * /dξ ∀c * , then (27) becomes
ρ o Sc = 1 0 |∇c|   1 0 ωc dc * |∇c||c *   dc |∇c| = 1 0 Ξ   +∞ -∞ ωc dξ   dc = ρ o S o L Ξ , (28)
where Eqs. ( 4) and ( 5) have been used and where it is also assumed that Ξ weakly varies with c. This last relation (Eq. ( 28)) which directly relates the turbulent consumption speed to the flame wrinkling will be useful to explicitly introduce the turbulent flame brush thickness in the analysis.

Scaling of turbulent burning velocity with flame brush thickness variation rate

To relate the flame brush thickness and the displacement speeds, the mass conservation and the progress variable equation (Eq. ( 7)) may be integrated through the turbulent reaction zones.

Still denoting ξ the coordinate in the direction normal to the reaction zone (dξ = dc/|∇c|) and with the boundary condition c = 0, ρ = ρ o for ξ → -∞ and c = 1 for ξ → +∞,

∂ ∂t +∞ -∞ ρdξ + (ρũ) +∞ = (ρũ) -∞ = ρ o S T , (29)
∂ ∂t

+∞ -∞ ρcdξ + (ρũ) ∞ = +∞ -∞ ωc dξ = ρ o Sc . ( 30
)
From these relations

S T -Sc = ∂ ∂t +∞ -∞ ρ ρ o (1 -c) dξ . (31)
Let us change variable introducing η(t) = ξ/δ T (t), where δ T (t) is the mean flame brush thickness. Then, for every variable φ(ξ, t) = φ(η(t))

∂ ∂t

ξ 2 ξ 1 φ(ξ, t)dξ = η 2 η 1 dφ(η) dη dη dt δ T (t)dη = η 2 η 1 dφ(η) dη - η δ T (t) dδ T (t) dt δ T (t)dη , = - dδ T (t) dt η 2 η 1 dφ dη ηdη = - dδ T (t) dt   [φη] η 2 η 1 - η 2 η 1 φdη   . (32)
Applied to the integral term in [START_REF] Cifuentes | Local volumetric dilatation rate and scalar geometries in a premixed methane-air turbulent jet flame[END_REF] this relation returns

∂ ∂t +∞ -∞ ρ ρ o (1 -c) dξ = dδ T (t) dt η + η - ρ(η) ρ o (1 -c(η)) dη I 1 >0 , (33)
where η ± (η -≈ 0 and η + ≈ 1) denote the cold and hot edges of the mean flame brush. The relation [START_REF] Cifuentes | Local volumetric dilatation rate and scalar geometries in a premixed methane-air turbulent jet flame[END_REF] with ( 33) and ( 28) becomes

S T = Sc + I 1 dδ T dt = S o L Ξ + I 1 dδ T (t) dt . (34)
Therefore, even under the hypothesis of thin and unstrained premixed flamelets, the turbulent flame speed can loose its direct linear proportionality to the flame area, as the flame brush thickness increases or decreases (i.e., for dδ T (t)/dt = 0).

A governing equation for the flame brush thickness δ T (t) has been discussed from first principles [START_REF] Peters | Turbulent Combustion[END_REF][START_REF] Oberlack | On symmetries and averaging of the G-equation for premixed turbulent combustion[END_REF][START_REF] Wenzel | Direct numerical simulation and modeling of kinematic restoration, dissipation and gas expansion effects of premixed flames in homogeneous turbulence[END_REF] leading to a scaling law for its relaxation starting from an initial condition δ T (t o ) within a turbulence featuring T and τ T as turbulent integral length and time scales, respectively,

δ 2 T (t) = b 2 1 2 T (1 -exp(-b 2 (t -t o )/τ T )) + δ 2 T (t o ) exp(-b 2 (t -t o )/τ T ) , (35)
where b 1 and b 2 are two parameters determined by expected asymptotic behaviors [START_REF] Peters | Turbulent Combustion[END_REF] or DNS [START_REF] Wenzel | Direct numerical simulation and modeling of kinematic restoration, dissipation and gas expansion effects of premixed flames in homogeneous turbulence[END_REF]. The relation (35) suggests a relaxation of the turbulent flame brush thickness following an exponential behaviour towards a value proportional to T . The time variation of δ T (t) is therefore controlled by the distribution of T and τ T encountered by the reaction zones, with dδ T (t)/dt rapidly evolving for the short time-scales (t -t o < τ T ), when the flame brush thickness is submitted to a non-uniform turbulence environment.

According to Eq. [START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated manifolds using convolutional neural networks[END_REF] this can lead to a quite large variety of responses of S T versus the integral turbulent length scale, depending on whether it is driven mainly by S o L Ξ or by I 1 dδ T (t)/dt, or again by an equilibrium between both of them.

We cannot investigate in detail this specific point from the DNS discussed above because it is limited to a streamwise length of one burner diameter, hence with an almost uniform δ T over the DNS zone. In fact most standard DNS suffers from such limitation. However, the link between T and S T was recently thoroughly discussed in [START_REF] Attili | Turbulent flame speed and reaction layer thicknening in premixed jet flames at constant Karlovitz and increasing Reynolds numbers[END_REF] to conclude that S T can increase exponentially with the integral scale, for T up to about 6 laminar flame thicknesses, evolution followed by a power-law scaling for larger values of T . This was observed together with S T loosing its proportionality to the flame surface area as the integral length scale increases, as expected from (34) combined with (35).

The very same analysis can be pursued in a frame moving with the mean flame brush and by integrating the continuity equation from the cold edge of the flame up to a point η = ξ /δ T (t) located between fresh and burnt gases:

∂ ∂t ξ -∞ ρdξ + ρ V T (ξ ) = ρ o S T , (36)
ρ o dδ T (t) dt η η - -η d(ρ/ρ o ) dη dη I 2 (η )>0 +ρ V T (η ) = ρ o S T . (37)
Combined with Eq. ( 34) and using the fact that through the turbulent flame η = η(c),

ρ(c) V T (c) = ρ o S o L Ξ + ρ o (I 1 -I 2 (c)) dδ T (t) dt . (38)
Where for c → 0, I 2 (c) → 0 and the cold leading edge relation ρ o S T = ρ o V T (c → 0) is recovered. Equation [START_REF] Vreman | An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications[END_REF] reveals that, within the turbulent flame brush, the displacement speeds of the iso-c surfaces are not simply proportional to the leading edge turbulent flame speed, but varies according to the dynamics of the flame brush thickness itself. Which according to (35) directly relates to the local turbulence properties and specifically to their eventual relaxation with time, making the modeling of S T from global turbulence quantities quite challenging.

Finally, the combination of ( 23) with [START_REF] Vreman | An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications[END_REF] illustrates the intricate links between turbulent transport, flame wrinkling and the variation of the flame brush thickness.

∇ • τ c = ρ o |∇c| (1 + τ ) ρ ρ o 2 -1 S o L Ξ -(I 1 -I 2 (c)) dδ T (t) dt . (39)
This last relation goes along with the well-known dependence of the gradient or counter gradient behaviors of the scalar turbulent transport on the dynamics of the turbulent flame brush thickness [START_REF] Libby | Countergradient diffusion in premixed turbulent flames[END_REF][START_REF] Veynante | Gradient and counter-gradient scalar transport in turbulent premixed flames[END_REF].

Conclusion

A relation between the variation rate of the turbulent flame brush thickness, the flame wrinkling and the turbulent burning velocity is established from the progress variable balance equation combined with the Bray-Moss-Libby (BML) theory. This relation derives from exact expressions for the relative progression velocities widely used in premixed turbulent flame analysis, combined with a flamelet approximation. From direct numerical simulation (DNS), it is shown that this essentiel ingredient of the derivation (i.e., the implication of the thin hypothesis on the statistics) stays valid, even when quite strong departures from the flamelet behavior are observed locally.

The role of the time derivative of the turbulent flame brush thickness in the expression for the turbulent burning velocity may explain previously reported deviation from the direct scaling between the flame surface amplification by turbulence, the integral properties of the turbulence (integral length and time scales) and the overall burning velocity. It is also discussed how the dynamics is likely to be driven by a short time scale, which is representative of the relaxation of the turbulent flame brush towards the integral length scale. Because of the large variability in the time evolution of the turbulence properties, this suggests in return the possibility of a large range of scalings for the turbulent burning velocity.

The development in a near future of DNS databases featuring premixed turbulent flames developing in shear flows over sufficiently large distances and at Reynolds number large enough, should allow for further investigating the obtained relations.

Figure 1 :

 1 Figure 1: Three LES-DNS zones of the jet-flame simulation. Mesh and iso-progress variable c = 0.8. h denotes the mesh resolution.

Figure 2 :

 2 Figure 2: Magnitude of the gradient of the progress variable. Scatter plot: |∇c| DNS (0.5% of the points shown); Square: |∇c| | c * ; Solid line: G(c) = |∇c| 1D-flamelet.

Figure 3 :

 3 Figure 3: Histogram of ρV r /(ρ o S o L ) from Eq. (3).

Figure 4 :Figure 5 :

 45 Figure 4: (a)-(c): various terms contributing to V r (Eq. (3)); Gradients in m -1 . Source term in kg • m -3 • s -1 . (d): G(c) gradient mapped from 1D reference flamelet. (e): Normalised relative progression velocity at flamelet-like locations (lenghts in m).

Figure 6 :

 6 Figure 6: ρ V T |c . LES filtering (∆ = 600 and 1200 µm). (a)-(b) all DNS points. (c)-(d): only points featuring 1D flamelet gradient. Solid line: Eq. (11). Circle: Thin flamelet approximation (Eq. (16)). Dashed line: Scalar sub-grid turbulent transport ∇ • τ c |c .

Acknowledgments

The authors thanks Prof. Ken Bray and Shirley Bray for 30 years of warm friendship.

The authors benefited from multiple fruitful collaborations with Prof. Ken Bray on reacting flow physics, including those discussed in this article.