Anja Kohl

The b-chromatic number of powers of cycles

Keywords: b-chromatic number, coloring, b-coloring, powers of cycles

A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k -1 color classes. The b-chromatic number χ b (G) is the maximum integer k for which G has a b-coloring by k colors. Let C r n be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χ b (C r n ) for all values of n and r, except for 2r + 3 ≤ n ≤ 3r. For the missing cases they presented the lower bound L := min{n -r -1, r + 1 + n-r-1 3 } and conjectured that χ b (C r n ) = L. In this paper, we determine the exact value on χ b (C r n ) for the missing cases. It turns out that χ b (C r n ) > L for 2r + 3 ≤ n ≤ 2r + 3 + r-6 4 .

Introduction

Let G = (V, E) be a simple, undirected graph with vertex set V and edge set E. For x, y ∈ V we denote by d(x, y) the distance between x and y, which is the number of edges in a shortest (x, y)-path. The rth power of a graph G, written as G r , is a graph on the same vertex set such that two vertices are joined by an edge if and only if their distance in G is at most r. For r ≥ 1, let C r n and P r n denote the rth power of a cycle and a path on n vertices, respectively.

A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a vertex having neighbors in all other k -1 color classes. Such a vertex is called a color-dominating vertex. The b-chromatic number χ b (G) is the maximum integer k for which G has a b-coloring by k colors. The concept of b-colorings and the b-chromatic number were introduced by Irving and Manlove [START_REF] Irving | The b-chromatic number of a graph[END_REF] in 1999 and have already been investigated in more than 50 papers (cf. [START_REF] Alkhateeb | Upper bounds on the b-chromatic number and results for restricted graph classes[END_REF]- [START_REF] Kratochvíl | On the b-chromatic number of graphs[END_REF]).

Since the problem to decide whether χ b (G) ≥ K for a given graph G and an integer K is N P-hard in general ( [START_REF] Irving | The b-chromatic number of a graph[END_REF]), we are interested in exact values on the b-chromatic number for special graphs or graph classes. In 2003, Effantin and Kheddouci [START_REF] Effantin | The b-chromatic number of some power graphs[END_REF] determined χ b (P r n ) for all values of n and r. They also investigated powers of cycles C r n and obtained the following result:

if 3r + 1 ≤ n ≤ 4r, 2r + 1 if n ≥ 4r + 1, (1)
and χ b (C r n ) ≥ min{n -r -1, r + 1 + n-r-1

3 } if 2r + 3 ≤ n ≤ 3r . (2)
The determination of χ b (C r n ) for 2r + 3 ≤ n ≤ 3r was posed as an open problem, but the authors conjectured that the lower bound in [START_REF] Blidia | On b-colorings in regular graphs[END_REF] is the exact value.

In the present paper, we complete Effantin and Kheddouci's work by determining χ b (C r n ) for the missing cases 2r + 3 ≤ n ≤ 3r. It turns out that the lower bound in [START_REF] Blidia | On b-colorings in regular graphs[END_REF] is indeed the exact value on χ b (C r n ) for 2r + 3 + r-6 4 < n ≤ 3r, but not for 2r + 3 ≤ n ≤ 2r + 3 + r-6 4 .

Main Result

Theorem 2 Let C r n be the rth power of a cycle of order n with 2r + 3 ≤ n ≤ 3r. Moreover, set l := n-2(r +1), R := r +1+3l mod (5l), and m := max R-2l

3 , 0 . Then, χ b (C r n ) = 3n-R 5 +m.

Preliminaries

In the following, we only consider powers of cycles C r n for which 2r + 3 ≤ n ≤ 3r is satisfied. Let V be the vertex set of C r n and set l := n -2(r + 1). Then, the complement of C r n is an (l + 1)-regular graph and

1 ≤ l ≤ r -2. Moreover, C r n has independence number 2. Let c be a b-coloring of C r n by k := χ b (C r n ) colors and let V 1 , . . . , V k be the corresponding color classes such that 1 ≤ |V 1 | ≤ |V 2 | ≤ . . . ≤ |V k | ≤ 2.
By a we denote the number of color classes of cardinality 1. Choose a color-dominating vertex v i ∈ V i for i = 1, . . . , k. Let A := {v 1 , . . . , v a }, B := {v a+1 , . . . , v k }, and C := {w a+1 , . . . , w k } where w i is the partner of v i , i.e. the vertex in V i \{v i }, for i ∈ {a+1, . . . , k}. Vertices from the set A, B, and C are called A-, B-, and C-vertices, respectively.

A vertex-row X shall be a set of vertices from V which are consecutive on the underlying cycle C n . If X only consists of (non-)A-vertices, then we call it a (non-)A-vertex-row (analogously for B and C).

The neighborhood N (x) of a vertex x ∈ V is the set of neighbors of x. Moreover, let N (x) := V \ (N (x) ∪ {x}) be the non-neighborhood of x and for a set

V ⊆ V let N (V ) := x∈V N (x). Note that |N (X)| = l + |X| for every vertex-row X. For a vertex u ∈ V and a set V ⊆ V let d Cn (u, V ) := min{d Cn (u, v) | v ∈ V } where d Cn (u, v) denotes the distance of u and v in the underlying C n .
We refer to Figure 1 for the following considerations. Let X be an A-vertex-row with α ≥ 1 vertices. We denote the two subgraphs induced by the vertex-rows between X and N (X) in clockwise order and in anti-clockwise order (on the underlying C n ) by G r (X) and G l (X), respectively (compare left side of Figure 1). From above we know that N (X) contains exactly l + α vertices. Moreover, the set N (N (X)) \ X has then exactly 2l vertices, and by symmetry we obtain

|N (N (X)) ∩ V (G l (X))| = |N (N (X)) ∩ V (G r (X))| = l.
Since each vertex from N (X) is non-adjacent to at least one A-vertex (which corresponds to a color class of cardinality 1), it cannot be color-dominating. So, each vertex from N (X) is a C-vertex and therefore has a B-vertex as partner. Let N p (X) be the set of partners of the vertices from N (X). Clearly, N p (X) ⊆ N (N (X)) \ X.

v l (X) vr(X) vertex from X → A-vertex vertex from N p (X) → B-vertex vertex from N (X) → C-vertex vertex from N (X) \ N (X) → non-A-vertex α l + α N (X) β l (X) + βr(X) = l + α ≥ β l (X) ≥ βr (X) ≥ βr (X) ≥ β l (X)
Fig. 1: Non-neighborhood N (X) and extended non-neighborhood N (X) of an A-vertex-row X Let v l (X) be the vertex from N p (X) ∩ V (G l (X)) that have the largest distance to the set

X in C n , i.e. d Cn (v l (X), X) = max v∈N p (X)∩V (G l (X)) d Cn (v, X). Moreover, set β l (X) := |N p (X) ∩ V (G l (X))|.
Analogously, we define v r (X) and β r (X). From above we know that

β l (X) ≤ d Cn (v l (X), X) ≤ |N (N (X)) ∩ V (G l (X))| = l and, analogously, β r (X) ≤ d Cn (v r (X), X) ≤ l. Along with β l (X) + β r (X) = |N p (X)| = |N (X)| = l + α ≥ l + 1, we further obtain β l (X), β r (X) ≥ 1.
This implies that both vertices v l (X) and v r (X) exist.

Since the partners of v l (X) and v r (X) are in N (X), we know that the sets N (v l (X)) and N (X) as well as the sets N (v r (X)) and N (X) intersect, and we can determine that |N

(v l (X)) \ N (X)| = d Cn (v l (X), X) and |N (v r (X)) \ N (X)| = d Cn (v r (X), X). Since each vertex from N (v l (X)
) is nonadjacent to the color-dominating vertex v l (X), it cannot be an A-vertex. The same holds for the vertices from N (v r (X)). So, altogether,

N (v l (X)) ∪ N (X) ∪ N (v r (X)) is a non-A-vertex-row with l + α + d Cn (v l (X), X) + d Cn (v r (X), X) ≥ l + α + β l (X) + β r (X) = 2(l + α) vertices.
We call this set the extended non-neighborhood of X and denote it by N (X). If X consists of only one vertex x, then we just write N (x) instead of N ({x}). The same we do with all previously defined notations.

Observation 1 Let X 1 , . . . , X t be a family of t ≥ 1 disjoint A-vertex-rows (in clockwise order on the C n ) such that N (X i ) ∩ N (X i+1 ) = ∅ for i ∈ {1, . . . , t -1}. Moreover, we assume the given A-vertex-rows to be maximal, i.e. each of it is delimited by non-A-vertices. Then,

t i=1 |X i | ≤ l.
Proof: Let Y i be the non-A-vertex-row between X i and X i+1 for i ∈ {1, . . . , t -1} and let Y 0 and Y t be the vertex-rows with l vertices preceding X 1 and succeeding X t , respectively. Moreover, set X :

= t i=1 X i and Y := t-1 i=1 Y i . Note that X ∪ Y is a vertex-row and therefore satisfies |N (X ∪ Y)| = l + |X ∪ Y|. Since N (X i ) and N (X i+1 ) intersect for each i ∈ {1, . . . , t -1}, it follows that N (Y i ) ⊂ (N (X i ) ∪ N (X i+1 )) and therefore N (Y) ⊂ N (X ). Moreover, it implies that N (X ) is a C-vertex-row.
For its cardinality we obtain:

Anja Kohl |N p (X )| = |N (X )| = |N (X ) ∪ N (Y)| = |N (X ∪ Y)| = l + |X ∪ Y| = l + t i=1 X i ∪ t-1 i=1 Y i = l + t i=1 X i + t-1 i=1 Y i = l + t i=1 |X i | + t-1 i=1 |Y i |. (3)
Moreover, we can easily check that

N (N (X )) = t i=1 X i ∪ t i=0 Y i = X ∪ t i=0 Y i . Since N p (X ) ⊆ N (N (X )) \ X , we then obtain: |N p (X )| ≤ |N (N (X )) \ X | = t i=0 Y i ≤ t i=0 |Y i | = 2l + t-1 i=1 |Y i |. (4)
So Inequalities ( 3) and (4) yield

t i=1 |X i | ≤ l. 2
Note that for t = 1, the previous observation yields that every A-vertex-row contains at most l vertices.

Proof of Theorem 2

Recall that k = χ b (C r n ), a = |A|, and l = n -2(r + 1). Since the b-coloring by k colors partitions the vertex set into a color classes of cardinality 1 and k -a color classes of cardinality 2, we obtain n = a + 2(k -a) and, thus, a = 2k -n = 2k -(l + 2r + 2) = 2(k -r -1) -l. So, a, l, and n have the same parity. Moreover, according to Inequality ( 2

), k ≥ min{n -(r + 1), r + 1 + n-(r+1) 3 } = min{l + r + 1, r + 1 + l+r+1 3 } = r + 1 + min{l, l+r+1 3 } and therefore a + l = 2k -n + l = 2k -2(r + 1) ≥ 2 min l, l + r + 1 3 . (5)
Let Q and R be the quotient and the remainder of the integer division r + 1 + 3l by 5l, i.e. r + 1 + 3l = Q • 5l + R, where Q ≥ 0 and 0 ≤ R < 5l. Note that then Q = r+1+3l 5l , R = r + 1 + 3l mod (5l), and n = l + 2(r + 1) = (2Q -1)5l + 2R. Moreover, set m := max R-2l 3 , 0 .

Case 1. l + 3 ≤ r + 1 < 2l, i.e. 5(r+1) 2 n ≤ 3r. At first we notice that this case is only possible for l ≥ 4. Moreover, by [START_REF] Irving | The b-chromatic number of a graph[END_REF] it follows that

a + l ≥ 2 l + r + 1 3 ≥ 2(l + r -1) 3 = 2r -(l + 2) 3 + l ≥ l + 2 3 + l ≥ 2 + l. (6)
Hence, there exist at least two A-vertices. Choose two vertices u 1 , u 2 ∈ A of maximum distance in C n . W.l.o.g. u 1 and u 2 are ordered in such a way, that a shortest (u 1 , u 2 )-path in C n runs anti-clockwise.

Then there are two cases to consider depending on the positions of u 2 and v l (u 1 ) (compare Figure 2). From the preliminaries and the premise of the case, we know that the extended non-neighborhood of u 1 has cardinality | N (u 1 )| ≥ 2(l + 1) ≥ r + 4. So, since N (u 1 ) is a vertex-row of at least r + 4 vertices, but the maximal vertex-rows induced by N (u 2 ) have only r vertices, we deduce that u 2 cannot be adjacent to all vertices in N (u 1 ), and so N (u 1 ) and N (u 2 ) intersect. Hence, N (u 1 ) ∪ N (u 2 ) is a non-A-vertex-row. Fig. 2: Extended non-neighborhood N (u1) and non-neighborhood N (u2) intersect Subcase 1.1. u 2 and v l (u 1 ) are in clockwise order (see left side of Figure 2). Since N (u 1 )∪N (u 2 ) is a non-A-vertex-row and by the choice of u 1 and u 2 as two A-vertices of maximum distance, there is no A-vertex in V (G l (u 2 )) ∪ N (u 2 ) (below the double-line). The remaining vertex set V (G r (u 2 )) ∪ {u 2 } (above the double-line) induces a vertex-row with exactly r + 1 vertices. This vertexrow has to contain all a A-vertices and all l + 1 B-vertices from N p (u 1 ). Hence, r + 1 ≥ a + l + 1. This and the premise of the case yield:

a + l ≤ r = (r + 2) + 2(r + 1) -4 3 ≤ 2l + 2(r + 1) -4 3 ≤ 2 l + r + 1 3 . (7)
Along with Inequality (6), we obtain a + l = 2 l+r+1 3 .

Subcase 1.2. u 2 and v l (u 1 ) are in anti-clockwise order (see right side of Figure 2). By the choice of u 1 and u 2 , there is no A-vertex in V (G l (u 2 )) ∪ N (u 2 ) ∪ V (G r (u 1 )) ∪ N (u 1 ) (below the double-curve). Hence, the remaining vertex set (V (G r (u 2 )) ∩ V (G l (u 1 ))) ∪ {u 1 , u 2 } (above the double-curve) that induces a vertex-row with 1 + d Cn (u 1 , u 2 ) vertices has to contain all a A-vertices. Since d Cn (u 1 , u 2 ) ≤ d Cn (u 1 , v l (u 1 )) -1 ≤ l -1, the non-neighborhoods N (u 1 ) and N (u 2 ) intersect. This implies that N (A) = N (u 1 ) ∪ N (u 2 ) and therefore

|N (A)| = l + 1 + d Cn (u 1 , u 2 ) ≥ l + a. Since N (A) ⊆ C, N p (A) ⊆ B, and |N (A)| = |N p (A)|, we deduce that n = |A| + |B| + |C| ≥ a + |N p (A)| + |N (A)| = a + 2 • |N (A)| ≥ a + 2(l + a) = 3a + 2l. Along with n = l + 2(r + 1) we then obtain a + l ≤ 2(l+r+1)

3

. This and Inequality (6) yield

2 l + r + 1 3 ≤ a + l ≤ 2(l + r + 1) 3 , (8)
and since a and l have the same parity implying that a + l is even, we conclude that a + l = 2 l+(r+1)

3

.

Hence, in both subcases we obtain a+l = 2 l+r+1 3 which implies that k = r+1+ l+r+1

3

. Moreover, since r + 1 + 3l < 5l by the premise of the case, it follows that R = r + 1 + 3l and m = r+1+l ). This implies that m = 0. We color the vertices from R l by colors (2Q -1)3l + 1, . . . , (2Q -1)3l + R in clockwise order. The same we do with the vertices from R r . Altogether, we obtain a coloring

3 . Therefore, k = r + 1 + l+r+1 3 = r + 1 + m = 3(l+2(r+1))-(r+1+3l) 5 + m = 3n-R 5 + m. Case 2. r + 1 ≥ 2l ≥ 2, i.e. 2r + 3 ≤ n ≤ 5(r+1) 2 . The premise yields r + 1 + 3l = Q • 5l + R ≥ 5l
                                  ❃ ⑥ ✇ ✴ R ′ l R ′ r R l Rr X2Q X2Q+1 N (X2Q+1) N (
c b of C r n by (2Q -1)3l + R = k colors. Subcase 2.2. 2(l + 1) < R < 5l.
Here, m > 0. We partition R l ∪ R r as depicted in Figure 4 into two bundles B 2Q , B 2Q+1 of 2l + 3m vertices each and two vertex-rows R l and R r of R -2l -3m vertices each. For i ∈ {2Q, 2Q + 1}, the bundle B i shall consist of a vertex-row X i with m black vertices, the non-neighborhood N (X i ) with l+m grey vertices, and N p (X i ) which is a subset of l + m black-white vertices from the set N (N (X i )) \ X i . Now we color the vertices from X 2Q in clockwise order by colors (2Q -1)3l + 1, . . . , (2Q -1)3l + m. Then we color the vertices from N (X 2Q ) as well as from N p (X 2Q ) in clockwise order by colors (2Q -1)3l + m + 1, . . . , (2Q -1)3l + 2m + l. Analogously, we color the vertices from X 2Q+1 , N (X 2Q+1 ), and N p (X 2Q+1 ) by colors (2Q -1)3l + 2m + l + 1, . . . , (2Q -1)3l + 4m + 2l. Finally, we color the vertices from R l as well as from R r in clockwise order by colors (2Q -1)3l + 4m + 2l + 1, . . . , (2Q -1)3l + 4m + 2l + (R -2l -3m) = (2Q -1)3l + R + m (compare the right side of Figure 5 for the case l = Q = 2, m = 1). Altogether, this yields a coloring c b of C r n by (2Q -1)3l + R + m = k colors.

                                     ✛ ... .
• • • • • • Fig. 5: b-coloring by k = (2Q -1)3l + R + m colors (here for l = Q = 2, m = 1)
It remains to prove that c b is a b-coloring for both subcases. Consider the Figures 345: The black vertices correspond to the color classes of cardinality 1 in the coloring c b . Moreover, each color class of cardinality 2 consists of a black-white and a grey vertex. Note that in Subcase 2.1 (resp. 2.2) we assume the vertices in R l (resp. R l ) to be black-white and the vertices in R r (resp. R r ) to be grey.

Because C r n has independence number 2, a vertex x from a color class V j has at least one neighbor in each color class V g of cardinality 2 for j = g. Hence, x is color-dominating if and only if x is adjacent to each vertex in a color class V h of cardinality 1 for j = h, i.e. if x is adjacent to all (other) black vertices.

Along with the fact that the black vertices are the vertices from s i=1 X i (s := 2Q -1 in Subcase 2.1 and s := 2Q + 1 in Subcase 2.2) this implies that x is color-dominating if and only if x / ∈ N (

s i=1 X i ) = s i=1 N (X i ).
Since all vertices from s i=1 N (X i ) are grey, we deduce that the black and the black-white vertices are color-dominating and correspond to the Aand the B-vertices, respectively. The grey vertices are then the C-vertices. Moreover, since every color class contains either a black or a black-white vertex it follows that every color class has a color-dominating vertex. Hence, c b is a proper b-coloring of C r n . For the number k of colors used in c b we obtain:

k = (2Q -1)3l + R + m = r + 1 + Ql + m = 5(r + 1) + (r + 1 + 3l -R) 5 + m = 3(l + 2(r + 1)) -R 5 + m = 3n -R 5 + m.
This implies that k ≥ k and because of a = 2k -n we further deduce that a ≥ 2k -n = 2(r + 1 + Ql + m) -(l + 2(r + 1)) = (2Q -1)l + 2m.

[" ≤ "] Now we prove that k ≤ r + 1 + Ql + m = 3n-R 5 + m. Consider the b-coloring c of C r n by k colors and suppose that k > k . This is equivalent to a > (2Q -1)l + 2m. Since a and l have the same parity, it suffices to consider a ≥ (2Q -1)l + 2(m + 1). Choose an arbitrary A-vertex-row X with α, 1 ≤ α ≤ l, vertices. We refer to Figure 1 for the following considerations.

Let β l (X) and β r (X) be defined as before. In the following, we abbreviate these values by β l and β r . Moreover, let a andc

l := |A ∩ V (G l (X))|, b l := |(B \ N p (X)) ∩ V (G l (X))|,
l := |C ∩ V (G l (X))|.
Analogously, we define a r , b r , and c r for the subgraph G r (X). Since G l (X) and G r (X) have the same order, we deduce that |V (G l (X))| = β l + a l + b l + c l = |V (G r (X))| = β r + a r + b r + c r . Every vertex from the set (B \ N p (X)) ∩ V (G l (X)) has a partner in C ∩ V (G r (X)), and vice versa. Hence, b l = c r . Analogously, we obtain b r = c l . Moreover, β l + β r = |N p (X)| = |N (X)| = l + α and a = α + a l + a r . Therefore, β l +a l = β r +a r = (l +α-β l )+(a-α-a l ) and altogether, 2(β l +a l ) = 2(β r +a r ) = l +a. By the assumption a ≥ (2Q-1)l+2(m+1) we obtain 2(β l +a l ) = 2(β r +a r ) = l+a ≥ 2Ql+2(m+1) and thus,

β l + a l = β r + a r ≥ Ql + m + 1. Moreover, since 1 ≤ β l , β r ≤ l (compare the preliminaries), we deduce that Ql + m + 1 ≤ β l + a l ≤ l + a l which yields a l ≥ (Q -1)l + m + 1. Analogously, we obtain a r ≥ (Q -1)l + m + 1.
Let X 1 , . . . , X q be the family of all maximal A-vertex-rows in V (G l (X)) (in clockwise order). For i ∈ {1, . . . , q -1} merge the two sets X i and X i+1 if N (X i ) ∩ N (X i+1 ) = ∅. Continue doing this until the remaining sets, say X 1 , . . . , X q (in clockwise order), satisfy N (X i )∩N (X i+1 ) = ∅ for i ∈ {1, . . . , q -1}. Set α i := |X i | for i ∈ {1, . . . , q}. Obviously, q i=1 α i = a l . Moreover, from Observation 1 we know that α i ≤ l for i ∈ {1, . . . , q}. Hence, ql ≥ q i=1 α i = a l ≥ (Q -1)l + m + 1 ≥ (Q -1)l + 1 and therefore q ≥ Q.

Since every vertex from

q i=1 N (X i ) is a C-vertex from V (G r (X)) we further deduce that b l = c r ≥ | q i=1 N (X i )| = q i=1 |N (X i )| ≥ q i=1 (l + α i ) = a l + ql ≥ a l + Ql.
Analogously, we can show that b r = c l ≥ a r + Ql. For the number of vertices n we then obtain:

n = |X| + |N (X)| + |N p (X)| + a l + a r + b l + b r + c l + c r = α + 2(l + α) + (a l + a r ) + 2(b l + b r ) = a + 2(l + α) + 2(b l + b r ) ≥ a + 2(l + α) + 2(a r + a l + 2Ql) = 3a + 2l + 4Ql ≥ 3(2Q -1)l + 6(m + 1) + 2l + 4Ql = 10Ql -l + 6(m + 1) (9)

  N (u1) ∪ N (u2) → non-A-vertex non-A-vertex by the choice of u1 and u2

Fig. 4 :

 4 Fig. 4: Partition of the remaining 2R vertices for the case m > 0

3 ≥ m + 1 . 5 + m. 2 3 3 } 4 . 3 , 0 . 3 . 3 = r+1+l 3 . 5 + 5 + r+1+l 3 = r + 1 + r+1+l 3 = L. Case 2 . 4 < n ≤ 5 (r+1) 2 . 5 = 3 ( 5 = 5 + 5 + 5 + 5 + m ≥ 1 . 5 = 1 . 5 + 5 +

 3152343033355332452535555515155 Moreover, since n = (2Q -1)5l + 2R, Inequality (9) yields R ≥ 2l + 3(m + 1) and thereforeR-2l This is a contradiction to m = max R-2l 3 , 0 . Therefore, k ≤ k , and altogetherk = k = 3n-R Conclusion In [3], Effantin and Kheddouci conjectured that χ b (C r n ) = min{n -(r + 1), r + 1 + n-(r+1)3} for 2r + 3 ≤ n ≤ 3r. The following observation shows that this conjecture does not hold in general.Observation 2 Let C r n be the rth power of a cycle of order n with 2r + 3 ≤ n ≤ 3r, and setL := min{n -(r + 1), r + 1 + n-(r+1) . Then, χ b (C r n ) = L if n > 9r+6 4 and χ b (C r n ) > L if n ≤ 9r+6 Proof: Recall that l = n -2(r + 1), R = r + 1 + 3l mod (5l), Q = r+1+3l-R 5l, and m = max R-2l So we obtain L = r + 1 + min{l, r+1+l 3 }. Case 1. l + 3 ≤ r + 1 < 2l, i.e. 5(r+1) 2 < n ≤ 3r. Then, L = r + 1 + r+1+l Moreover, we obtain r + 1 + 3l < 5l which implies that R = r + 1 + 3l and m = R-2l Thus, Theorem 2 yields χ b (C r n ) = 3n-R m = 3(l+2(r+1))-(r+1+3l) 2l ≤ r + 1 < 4l + 3, i.e. 9r+6 Then, L = r + 1 + l. Furthermore, it follows that 5l ≤ r + 1 + 3l < 7l + 3 = 5l + (2l + 3) and therefore Q = 1 and 0 ≤ R = r + 1 -2l < 2l + 3. This yields m = 0. Due to Theorem 2, we then obtainχ b (C r n ) = 3n-R l+2(r+1))-(r+1-2l) r + 1 + l = L.Case 3. r + 1 ≥ 4l + 3 ≥ 7, i.e. 2r + 3 ≤ n ≤ 9r+6 4 . Then, L = r + 1 + l. Moreover, we deduce that r + 1 + 3l ≥ 7l + 3 = 5l + (2l + 3) which yieldsQ ≥ 1. According to Theorem 2, χ b (C r n ) = 3n-R m = r + 1 + l + r+1-2l-R m = L + r+1-2l-R m (note that r+1-2l-R5 is an integer). In order to prove χ b (C r n ) > L, we have to show that r+1-2l-R If m = 0, then R ≤ 2l + 2 and therefore r+1-2l-R If m > 0, then r+1-2l-R m = r+1-2l-(r+1+3l-5lQ) m = (Q -1)l + m ≥ 0 + m ≥ 1. 2

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . .	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  and therefore Q ≥ 1. ... ... ... ... ... ... ... ... ... ... ... ..

	. . .	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	. . . .

  .. ... ... ... ... ... ... ... ... ... ..

			18	3	4	1 2	5	6	9
		17								10
		16								11
		15								12			22																											24
												21																													25
												19																														23
												20																														26
	R l	. . .								. . .	Rr	k ′ 27	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	27 k ′
												26																														20
		12								15		25																													21
		11								16			24																											22
		8								13
		7	10	9	6	5 4	3	18	17	14		A-					B-						C-						B-or C-vertex

Acknowledgements

The author would like to thank the referees for helpful comments and suggestions.

Anja Kohl

Theorem 1 (Effantin and Kheddouci, [START_REF] Effantin | The b-chromatic number of some power graphs[END_REF]) The b-chromatic number of C r n for r ≥ 1 is

of 5l vertices each and 2R remaining vertices. For i ∈ {1, . . . , 2Q -1}, the bundle B i shall consist of a vertex-row X i with l black vertices, the non-neighborhood N (X i ) with 2l grey vertices, and the set N p (X i ) := N (N (X i ))\X i with 2l black-white vertices, such that the sets X 1 , . . . , X 2Q-1 are positioned on the cycle C n as described in Figure 3. In order to do this, we start with an arbitrary vertex-row with l vertices as X 1 (on top in the picture). Then the positions of N (X 1 ) and N p (X 1 ) are already specified. After this, we position X 2 (bottom left) and X 3 (bottom right) leaving out l vertices on each side of N (X 1 ). Then the positions of N (X 2 ) and N p (X 2 ) as well as N (X 3 ) and N p (X 3 ) are already specified. If Q ≥ 3, then we proceed by positioning X 4 (top left) and X 5 (top right) leaving out l vertices on the left side of N (X 3 ) and on the right side of N (X 2 ), respectively. Then also the positions of N (X 4 ) and N p (X 4 ) as well as N (X 5 ) and N p (X 5 ) are specified. If Q ≥ 4, then we position the remaining sets X 6 , . . . , X 2Q-1 in the same manner as we positioned X 2 -X 5 using always the order bottom left, bottom right, top left, top right. The 2R remaining vertices induce two vertex-rows with R vertices each which we denote by R l and R r . Now we color the l vertices from X 1 in clockwise order by colors 1, . . . , l. Then we color the 2l vertices from N (X 1 ) in clockwise order by colors l + 1, . . . , 3l, and the same we do for the 2l vertices from N (N (X 1 )) \ X 1 = N p (X 1 ). Now all vertices from B 1 are colored. Next we color the vertices from B 2 in an analogous way by colors 3l + 1, . . . , 6l, and we continue until B 2Q-1 whose vertices we color by colors (2Q -2)3l + 1, . . . , (2Q -1)3l. This coloring procedure yields a coloring of (2Q -1)5l = n -2R vertices by (2Q -1)3l colors (compare the left side of Figure 5 for the case l = Q = 2). For the coloring of the vertices from R l ∪ R r we have to distinguish the following two cases:


Ciara R Wigham

Nicolas Guichon

Nicolas Guichon In

In and out the frame: teacher gestures during desktop videoconferencing interactions

come   L'archive ouverte pluridisciplinaire

 Wigham, C.R. & Guichon, N. (2014). In and out the frame: teacher gestures during desktop videoconferencing interactions, CALICO 2014, 6-10 mai 2014, Athens Ohio. This presentation will focus on the production of gestures by teacher trainees when they are interacting online with distant language learners during a desktop videoconferencing interaction. Previous research has provided a mixed picture of the use of the webcam in pedagogical situations: while it is seen as a useful medium to provide complementary information in communicative breakdown (Buckett, Stringer and Datta, 1999), the limited access to visual cues provided by the webcam is felt as limiting (Zähner, Fauverge and Wong, 2000) or even disruptive (Chanier and Ciekanski, 2008). Because the constraints brought by the webcam shape online communication, research is needed to understand how teachers manage with their own image (paralinguistic cues) and develop strategies to adjust their teaching.

In line with the domain of Gesture Studies (McCafferty & Stam, 2008), the present qualitative study aims to understand how teacher trainees use the affordances of the webcam to produce nonverbal cues that may be useful for mutual comprehension. The data are comprised of (1) dynamic screenshots of 40-minute pedagogical interactions (what the learners can see on their screens), (2) films that were shot during the interactions (the context of the interaction invisible to the learners), and (3) stimulated recall sessions during which the same trainees were asked to explain their choices concerning framing and the production of paralinguistic cues (nods, gestures, smiles, etc.). The participants are three French teacher trainees who are being trained to teach online via a multimodal interface over a seven-week period via a telecollaborative project with Irish learners of French. This study will concentrate on the production of gestures that are in the frame and out of the frame and will try to determine the functions of both categories. We will then explore how future French teachers adapt their gestures and what their intentions are when doing so. We contend that attention to framing, that is what the online teacher chooses consciously or unconsciously to include in the restricted field of the webcam frame, is one crucial aspect of the contextualization process at work in an online synchronous interaction (Jones, 2004).


Eric Goncalvès
email: eric.goncalves@legi.grenoble-inp.fr

Modeling for non isothermal cavitation using four-equation models

Keywords: Cavitation, Homogeneous model, Mass transfer, Thermal effects

A compressible, two-phase, one-fluid solver has been developed to investigate the behaviour of cavitation models including thermodynamic effects. The code is composed by three conservation laws for mixture variables (mass, momentum and total energy) and a supplementary transport equation for the void ratio. Two formulations for the mass transfer between phases are studied. Numerical simulations are firstly performed on rarefaction cavitating problems in which the working fluid is hot water and freon R-114. A realistic turbulent Venturi case with freon R-114 is performed and comparisons are done between 3-and 4-equation models. A warming effect is highlighted downstream the cavitation pocket in the region of pressure recuperation.

Introduction

Cavitation is a significant engineering phenomenon that occurs in fluid machinery, fuel injectors, marine propellers, nozzles, underwater bodies, etc. In most cases, cavitation is an undesirable phenomenon, significantly degrading performance, resulting in reduced flow rates, lower pressure increases in pumps, load asymmetry, vibrations, noise and erosion. Such flows are characterized by important variations of the local Mach number (due to the drastic diminution of the speed of sound in the mixture), large density ratio between the liquid and the vapor phases, compressibility effects and non equilibrium thermodynamic states.

Cavitation can be manifested at a constant temperature, and thus, it is usually assumed to be an isothermal phenomenon. However, the constant temperature hypothesis is no longer valid when cryogenic fluids (also known as thermosensitive fluids) are considered. For such fluids, the liquid-vapour density ratio is lower than that of typical fluids (cold water) and consequently more liquid mass has to vaporize to sustain a cavity. Therefore evaporative cooling effects are more pronounced and the temperature of the liquid in the immediate vicinity of the liquid-vapour interface is depressed below the free-stream temperature. Because of the strong variation of thermodynamic properties (vapour pressure, density), the temperature depression, negligible in water, is quite substantial. The local cooling effect delays the cavitation phenomenon and reduces the local vapour pressure of the fluid, which leads to a lower observed cavity pressure.

Several physical and numerical models have been developed to investigate cavitating flows within the framework of averaged two-phase model or bubbly models based on the Rayleigh-Plesset equation. For the averaged model, there are different approaches according to the assumptions made on the local thermodynamic equilibrium and the slip condition between phases. A hierarchy of models exists, with the numbers of equations ranging from seven to three only. The full non-equilibrium seven-equation models are the most complete. For both fluids, it contains equations for the mass, momentum and energy, and the seventh equation describes the topology of the flow.

These models can take into account the physical details occurring in the cavitation phenomenon such as mass exchange, thermal transfer and surface tension. However, the transfer terms have to be known; such quantities are usually very difficult to obtain. Various formulations have been investigated to deal with metastable states and evaporation front dynamics [START_REF] Saurel | A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation[END_REF][START_REF] Ishimoto | Numerical study of cavitating flow characteristics of liquid helium in a pipe[END_REF][START_REF] Metayer | Modelling evaporation fronts with reactive Riemann solvers[END_REF][START_REF] Yeom | Numerical simulation of two-fluid two-phase flows by HLL scheme using an approximate jacobian matrix[END_REF][START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF]. Temperature and free Gibbs enthalpy exchange terms are included in the equations as relaxation terms to model heat and mass transfer. For thermal-hydraulics applications with cavitation, nucleation and boiling flows, a six-equation model has been developed [START_REF] Mimouni | Modelling and computation of unsteady cavitation flows[END_REF][START_REF] Mimouni | Modelling and computation of cavitation and boiling bubbly flows with Neptune CFD code[END_REF]. The interfacial mass transfer is modeled as a function of the interfacial heat transfer terms and the inter-facial phase-averaged enthalpies.

A reduced five-equation model can be derived with the assumptions of velocity equilibrium and pressure equilibrium. The archetype five-equation model is that of Kapila [START_REF] Kapila | Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations[END_REF]. It is composed of four conservation laws: two for masses, one for the mixture momentum and one for the mixture energy. It is completed by an equation for a non-conservative quantity describing the flow topology, usually the void ratio. Such a model has been used for inviscid high speed cavitating applications and cavitation pocket in fuel injector nozzles [START_REF] Petitpas | A relaxation-projection method for compressible flows. Part II: artificial heat exchanges for multiphase shocks[END_REF][START_REF] Saurel | Modelling phase transition in metastable liquids: application to cavitating and flashing flows[END_REF]. Heat and mass transfer processes are modelled through thermal and chemical relaxation procedures.

By assuming the thermal equilibrium between phases, a four-equation model can be expressed. A very popular formulation, originally developed to simulate turbulent cavitating flows in cold water, has been adapted to cryogenic applications [START_REF] Hosangadi | Numerical study of cavitation in cryogenic fluids[END_REF][START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF][START_REF] Tseng | Modeling for isothermal and cryogenic cavitation[END_REF][START_REF] Zhang | Computational fluid dynamic study on cavitation in liquid nitrogen[END_REF]. It is composed by three conservation laws for mixture quantities (mass, momentum, energy) plus a mass equation for the vapour or liquid density including a cavitation source term. The main difficulty is related to the formulation of the source term and the tunable parameters involved for the vaporization and condensation processes (different sets of parameters are presented in [START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF]). Moreover, this family of models are not thermodynamically well-posed and does not respect thermodynamic constraints [START_REF] Goncalves | Constraints on equation of state for cavitating flows with thermodynamic effects[END_REF]. Another popular model devoted to ebullition problems uses a mass fraction equation with a relaxation term (Homogeneous Relaxation Model). The source term involves a relaxation time that is the time for the system to regain its thermodynamic equilibrium state. This time is difficult to determine and is estimated from experimental data [START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF][START_REF] Bilicki | Evaluation of the relaxation time of heat and mass exchange in the liquid-vapour bubble flow[END_REF][START_REF] Barret | Schemes to compute unsteady flashing flows[END_REF][START_REF] Schmidt | Multi-dimensional simulation of thermal non-equilibrium channel flow[END_REF].

With the assumption of complete thermodynamic equilibrium between phases (local temperature, pressure and free Gibbs enthalpy equality between phases), we obtain the 3-equation models or Homogeneous Equilibrium Models (HEM). Vaporization or condensation processes are assumed to be instantaneous. An equation of state (EOS) is necessary to close the system. Different closure relations (tabulated EOS or combination of pure phase EOSs) that link the pressure to the thermodynamic variables have been proposed [START_REF] Cooper | Analysis of single and two-phase flow in turbopump inducers[END_REF][START_REF] Clerc | Numerical simulation of the homogeneous equilibrium model for two-phase flows[END_REF][START_REF] Rapposelli | A barotropic cavitation model with thermodynamic effects[END_REF][START_REF] Rolland | Experiments and modelling of cavitating flows in venturi. Part I: stable cavitation[END_REF][START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF].

The bubbly flow models are composed by three balance equations for the mixture quantities coupled with a macroscopic model for the bubble dynamics based on the Rayleigh-Plesset equation. This model is capable of handling either single bubbles or clouds of bubbles that grow and decrease through a pressure field [START_REF] Fujikawa | Non-equilibrium vapour condensation on a shock-tube endwall behind a reflected shock wave[END_REF][START_REF] Franc | Analysis of thermal effects in a cavitating inducer using rayleigh equation[END_REF][START_REF] Rodio | Influence of convective heat transfer modeling on the estimation of thermal effects in cryogenic cavitating flows[END_REF]. In the case where heat transfer is negligible, the phase change is driven by inertia effects.

Yet, when thermal effects are involved, the liquid inertia become rapidly negligible and the evolution is controlled by the heat flux provided by the liquid at the bubble surface. By comparing characteristic times of thermal and inertial phenomena, it can be shown that this thermal regime is an accurate representation of reality for moderate levels of superheating or subcooling [START_REF] Florschuetz | On the mechanics of vapor bubble collapse[END_REF][START_REF] Prosperetti | Vapour-bubble growth in a superheated liquid[END_REF][START_REF] Okhotsimskii | The thermal regime of vapour bubble collapse at different jakob numbers[END_REF].

In a recent study, we proposed a new mass transfer formulation associated to a fourequation model for isothermal cavitation [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF][START_REF] Goncalves | Modelling for isothermal cavitation with a fourequation modek[END_REF]. The generic formulation involves the ratio c 2 /c 2 wallis between the mixture speed of sound and the Wallis velocity, which is the speed of sound without heat and mass transfer. First, we extend the isothermal formulation with a non isothermal thermodynamic path using a linear approximation of the vapour pressure evolution. This model is built using the mixture speed of sound evaluated with a modified barotropic equation of state [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF]. A second closure is investigated using a mixture of stiffened gas EOS and its associated mixture speed of sound. The validation is done through one-dimensional inviscid double rarefaction test cases in which reference solutions have been computed [START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF]. A new test case is proposed with the thermosensitive freon R-114 (C 2 Cl 2 F 4 ) as working fluid. Secondly, models are compared with experimental data on a turbulent Venturi case in which the running fluid is freon R-114. Local analyses with void ratio profiles and wall temperature depression are proposed. A warming effect downstream the cavitation pocket is exhibited. This paper is organized as follows. We give a brief description of models. The averaged Navier-Stokes equations are presented and the numerical methods are described.

Numerical results are presented with comparisons between models and validations against two-fluid solutions. The study of the turbulent venturi case is described.

Finally, conclusions and future investigations are discussed.

Mixture models and mass transfer

The numerical simulations are carried out using an in-house CFD code solving the one-fluid compressible Euler and Navier-Stokes systems.

The homogeneous mixture approach is used to model two-phase flows. The phases are assumed to be sufficiently well mixed and the disperse particle size are sufficiently small thereby eliminating any significant relative motion. The phases are strongly coupled and moving at the same velocity. In addition, the phases are assumed to be in thermal and mechanical equilibrium: they share the same temperature T and the same pressure P . The evolution of the two-phase flow can be described by the conservation laws that employ the representative flow properties as unknowns just as in a single-phase problem.

We introduce α the void fraction or the averaged fraction of presence of the vapour.

The density ρ, the center of mass velocity u and the internal energy e for the mixture are defined by [START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF]:

ρ = αρ v + (1 -α)ρ l (1) ρu = αρ v u + (1 -α)ρ l u (2) ρe = αρ v e v + (1 -α)ρ l e l ( 3
)
To close the system, an equation of state (EOS) is necessary to link the pressure and the temperature to the internal energy and the density. For the pure phases, we used the convex stiffened gas EOS (see [START_REF] Metayer | Elaborating equations of state of a liquid and its vapor for two-phase flow models[END_REF]):

P (ρ, e) = (γ -1)ρ(e -q) -γP ∞ (4) P (ρ, T ) = ρ(γ -1)C v T -P ∞ (5) T (ρ, h) = h -q C p (6)
where γ = C p /C v is the heat capacity ratio, C p and C v are thermal capacities, q the energy of the fluid at a given reference state and P ∞ is a constant reference pressure.

The speed of sound c is given by:

c 2 = γ P + P ∞ ρ = (γ -1)C p T (7)
For the mixture, two EOS are tested: a mixture of stiffened gas (SG) and a modified barotropic EOS.

A mixture of stiffened gas EOS

By assuming the pressure equilibrium between phases, an expression for the pressure can be deduced, function of the void ratio α and the vapour mass fraction Y = αρ v /ρ:

P (ρ, e, α, Y ) = (γ(α) -1)ρ(e -q(Y )) -γ(α)P ∞ (α) (8) 1 γ(α) -1 = α γ v -1 + 1 -α γ l -1 (9) q(Y ) = Y q v + (1 -Y )q l ( 10
)
P ∞ (α) = γ(α) -1 γ(α) [ α γ v γ v -1 P v ∞ + (1 -α) γ l γ l -1 P l ∞ ] (11)
By assuming the thermal equilibrium between phases, the mixture temperature is expressed as:

T (ρ, h, Y ) = h l -q l C p l = h v -q v C pv = h -q(Y ) C p (Y ) (12)
C p (Y ) = Y C pv + (1 -Y )C p l ( 13
)
We assume that the vaporization pressure varies linearly with the temperature:

P vap (T ) = P vap (T ref ) + dP dT (T -T ref ) (14)
The constant quantity dP/dT is evaluated with a thermodynamic table.

The speed of sound in the mixture can be expressed as a function of the enthalpy of each phase (see Appendix A):

C 1 = 1 γ -1 + ρ v h v -ρ l h l (ρ l -ρ v ) dP dT ( α dρ v dT + (1 -α) dρ l dT ) ( 15
)
ρc 2 = 1 C 1 [ ρ v ρ l (ρ l -ρ v ) (h v -h l ) ] (16)
Enthalpies of pure phase h l and h v are computed with the mixture temperature T .

Due to numerical problems, the derivatives of densities with the temperature are not taken into account in this study.

A modified barotropic EOS

A modified barotropic relation [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF] can be considered for the mixture. When the pressure is between P vap (T ) + ∆P and P vap (T ) -∆P , the following relationship applies:

P (α, T ) = P vap (T ) + ( ρ sat l -ρ sat v 2 ) c 2 baro Arcsin (1 -2α) ( 17
)
where ∆P represents the pressure range of the law and, for a void ratio value of 0.5, the pressure is equal to the saturation pressure P vap (T ) at the local temperature T .

This temperature is evaluated using the relation [START_REF] Tseng | Modeling for isothermal and cryogenic cavitation[END_REF]. ρ sat l and ρ sat v are pure phases densities at saturation evaluated at the reference temperature T ref . The quantity c baro , which has the dimension of a velocity, is a parameter of the model.

As previously we assume that the vaporization pressure varies linearly with the temperature and follows the relation [START_REF] Zhang | Computational fluid dynamic study on cavitation in liquid nitrogen[END_REF].

The speed of sound in the mixture can be written as [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF]:

c 2 = ρ V ρ L ρ(ρ L -ρ V ) (h V -h L ) dP dT + ρC p c 2 T ρC p -dP dT ( 18
)
C p = Y C pv + (1 -Y )C p l ( 19
)
c 2 T = ( ∂P ∂ρ ) s = ( ∂P ∂ρ ) T = c 2 baro 2 √ α(1 -α) ( 20
)
Where c T is the isothermal speed of sound (i.e. when dP/dT =0).

A void ratio transport-equation with mass transfer

The model consists in three conservation laws for mixture quantities and an additional equation for the void ratio. It is obtained from a reduction of the Kapila five-equation model assuming the liquid is at its saturation state [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF]. We present below only the void ratio equation:

∂α ∂t + V ∂α ∂x = ( ρ l c 2 l -ρ v c 2 v ρ l c 2 l 1-α + ρvc 2 v α ) = K div V + ( c 2 v α + c 2 l 1-α ρ l c 2 l 1-α + ρvc 2 v α ) = 1/ρ I the interf acial density ṁ ( 21
)
where ṁ denotes the mass transfer between phases, V the homogeneous velocity vector, c k the speed of sound of the phase k.

By assuming that the mass transfer is proportional to the velocity divergence, it is possible to build a family of models in which the mass transfer ṁ is expressed as

ṁ = ρ l ρ v ρ l -ρ v ( 1 - c 2 c 2 wallis ) div V ( 22
)
where c wallis is the propagation velocity of acoustic waves without mass transfer [START_REF] Wallis | One-dimensional two-phase flow[END_REF].

This speed of sound is expressed as a weighted harmonic mean of speeds of sound of each phase:

1 ρc 2 wallis = α ρ v c 2 v + 1 -α ρ l c 2 l ( 23
)
The liquid density ρ l is assumed to be in its equilibrium state at the reference temperature: ρ l = ρ sat l (T ref ). We did not test to introduce a thermal variation for the liquid density. The vapour density ρ v follows the pure phase EOS (i.e. the stiffened gas EOS) and varies with the temperature.

With this generic form for the mass transfer, we remark that all models in which the mixture speed of sound is the Wallis one can not produce or destroy the void ratio during the phase transition (it is the case of most of void ratio transport equation models). The void ratio is only modified through the term K involving the compressibility of pure phases.

Model with a barotropic EOS

A first model is built using the speed of sound of the modified barotropic model presented previously. In the following, this model will be named 4-equation barotropic model. It involves two parameters: c baro and dP/dT .

As the temperature and pressure relations are coupled (relations 14 and 17), an iterative procedure on the temperature is introduced. The temperature is initialized by the liquid temperature computed with the pure phase EOS. Five iterations are done and it has been checked that the numerical solution did not change with a higher number of iteration.

Without thermal variation, this model has been successfully tested on rarefaction tube problems [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF] and cavitating venturi flows [START_REF] Goncalves | Modelling for isothermal cavitation with a fourequation modek[END_REF].

Model with a mixture of stiffened gas EOS

A second model is built using the mixture of stiffened gas EOS. The mass transfer term is activated when the local pressure P is smaller than the vapour pressure P vap (T ) evaluated using the relation [START_REF] Zhang | Computational fluid dynamic study on cavitation in liquid nitrogen[END_REF]. This model will be named 4-equation SG model. It involves only the parameter dP/dT . As previously, the temperature and pressure relations are coupled and an iterative procedure on the temperature is done with five iterations. It has been checked that the numerical solution did not change with a higher number of iteration.

Characteristics of both four-equation models and the three-equation barotropic model [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF] are summarized in Table 8. In addition, characteristics of the two-fluid model of Zein et al. [START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF] based on relaxation procedures on the pressure, temperature and Gibbs enthalpy are presented.

Speed of sound in the mixture

The mass transfer formulation involves explicitly the difference between the speed of sound in the mixture and the Wallis one. Fig. 1 compares the evolution of this speed of sound (with a logarithmic scale) as a function of the void ratio for the EOS presented, the Wallis formulation and the equilibrium speed of sound in a mixture of freon R-114. Without heat and mass transfer effects, the propagation of acoustic waves follows the Wallis speed of sound. When exchanges of mass and heat between phases are involved, the sound speed decreases to the thermodynamic equilibrium one [START_REF] Petitpas | Diffuse interface model for high speed cavitating underwater systems[END_REF]. This limit speed is evaluated with the assumption of local thermodynamic equilibrium between phases: equalities of pressure, temperature and free Gibbs enthalpy between phases. The expression of this velocity is given in [START_REF] Goncalves | Constraints on equation of state for cavitating flows with thermodynamic effects[END_REF].

To compare the different speeds, the temperature is assumed to be constant equal to 293 K and saturation values are used. The parameters of the stiffened gas EOS are given in table 2. For the barotropic EOS, the parameter c baro is set at 1.5 m/s. We can observe that for both EOS the mixture speed of sound is well limited by the Wallis and the equilibrium ones. Both models are able to create vapour as soon as the divergence of velocity is strictly positive. Moreover, we remark that the barotropic speed of sound is very close to the equilibrium one while the SG speed is relatively close to the Wallis one.

Reynolds averaged Navier-Stokes equations

For turbulent computations, the compressible one-fluid RANS equations are used, coupled with the one-equation turbulence model of Spalart-Allmaras (SA) [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. These equations can be expressed as:

∂w ∂t + div (F c -F v ) = S (24) w =            ρ ρV ρE α ρν            ; F c =            ρV ρV ⊗ V + P I (ρE + P )V αV ρνV            ; F v =            0 τ v + τ t (τ v + τ t ).V -Q v -Q t 0 (µ + ρν/σ ν ) grad ν           
where w denotes the conservative variables and the void ratio, F c and F v the convective and viscous flux densities and S the source terms, which concern the void ratio equation and the turbulent transport equation. E = e + u 2 /2 is the mixture total energy. ν is the transported turbulent variable.

The total stress tensor τ is evaluated using the Stokes hypothesis, Newton's law and the Boussinesq assumption. The total heat flux vector Q is obtained from the Fourier law involving a turbulent thermal conductivity λ t with the constant Prandtl number hypothesis.

τ = τ v + τ t = (µ + µ t ) [ ( grad V + ( grad V ) t ) - 2 3 ( div V )I ] + 2 3 ρkI Q = Q v + Q t = -(λ + λ t ) grad T with λ t = µ t C p P rt ( 25
)
In the pure liquid, the viscosity is determined by an exponential law and, in pure vapour, the viscosity follows the Sutherland law. The mixture viscosity is defined as the arithmetic mean of the liquid and vapour viscosities (fluctuations of viscosity are neglected) [START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF]:

µ L (T ) = µ 0 L exp (B/T ) (26) µ V (T ) = µ 0 V √ T 293 1 + T S /293 1 + T S /T (27) µ(T, α) = αµ V (T ) + (1 -α)µ L (T ) (28)
where µ 0 L , µ 0 V , B and T S are constant parameters.

The mixture thermal conductivity λ is also defined as the arithmetic mean of the liquid and vapour values:

λ(α, T ) = α µ V (T )C p V P r V + (1 -α) µ L (T )C p L P r L ( 29
)
The turbulent Prandtl number P rt is set to 1.

For the modelling of flow close to the wall, a two-layer wall law approach is used [START_REF] Goncalves | Wall model and mesh influence study for partial cavities[END_REF].

Numerical methods

The numerical simulations are carried out using an explicit CFD code based on a finite-volume discretization. For the mean flow, the convective flux density vector on a cell face is computed with the Jameson-Schmidt-Turkel scheme [START_REF] Jameson | Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes[END_REF]. The artificial viscosity includes a second-order dissipation term D 2 and a fourth-order dissipation term D 4 , which involve two tunable parameters k (2) and k (4) .

The viscous terms are discretized by a second-order space-centered scheme. For the turbulence transport equations, the upwind Roe scheme [START_REF] Roe | Approximate Riemann solvers, parameters vectors, and difference schemes[END_REF] is used to obtain a more robust method. The second-order accuracy is obtained by introducing a flux-limited dissipation [START_REF] Tatsumi | Flux-limited schemes for the compressible Navier-Stokes equations[END_REF].

The numerical treatment of boundary conditions is based on the use of the characteristic relationships. More details are given in [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF].

1D inviscid rarefaction test case with water

A double rarefaction tube problem is considered with an initial velocity discontinuity located at the middle of the tube. This test consists in a one meter long tube filled with liquid water at atmospheric pressure and with density ρ l =1150 kg/m 3 . The The solution involves two expansion waves. As gas is present, the pressure cannot become negative. To maintain positive pressure, the gas volume fraction increases due to the gas mechanical expansion and creates a pocket. Liquid water is expanded until the saturation pressure is reached then evaporation appears and quite small amount of vapor is created. The solution with phase transition is composed of four expansion waves. The extra two expansion waves correspond to the evaporation fronts.

These cases were computed in [START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF] using a two-fluid model in which instantaneous relaxation processes toward equilibrium are included for the temperature and the Gibbs free energy. The mesh contains 5000 cells. The time step is set to 10 -7 s. The value of c baro is set to 1.31 m/s for the 4-equation barotropic model as presented in [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF].

The parameters of the stiffened gas EOS and saturation values for densities are given in Table 3. The quantities have been evaluated with a saturation table at the reference temperature.

Velocity | u |= 2 m/s

The value of dP/dT , evaluated with a thermodynamic table on the temperature in- Both models provide a similar evolution for the pressure, velocity, volume and mass fraction of gas, in good agreement with the two-fluid solution. The mixture speeds of sound for both models are compared with the Wallis formulation (with a logarithmic scale). A large discrepancy is noticeable between models. Using the SG model, the speed of sound is equal to the Wallis one excepted in the cavitation area where it is a little smaller. This small gap is enough to create the cavitation pocket. On the contrary, the speed of sound given by the barotropic model is clearly lower than the Wallis one (by a factor 5), as observed in Figure 1.

The temperature evolution puts in evidence the small cooling effect during the phase transition. For this test case, the thermal effect is weak. The temperature drop reaches 0.25 K with the 4-equation barotropic model and 0.5 K with the 4-equation SG model. Unfortunately the temperature evolution was not plotted in [START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF].

A simple heat balance between the two phases can estimate the scale of temperature difference ∆T * caused by thermal effects. The B-factor is estimated as the ratio between the actual temperature drop and ∆T * :

∆T * = ρ v L vap ρ l C p l and B = ∆T ∆T * ( 30
)
where L vap is the latent heat and C p L represents the specific heat.

The B-factor of ideal gas-liquid mixture is expressed by the following equation [START_REF] Franc | Fundamentals of cavitation[END_REF]:

B = α 1 -α ( 31
)
With the maximum value reached during the computation, that is α = 0.7, the ideal temperature depression is around 0.33 K. Both models provide a cooling effect close to this value.

Velocity | u |= 100 m/s

The same conditions are used except regarding velocities which are set to |u|=100 m/s. In this case, evaporation is much more intense resulting in a large cavitation pocket where the gas volume fraction is close to 1. However, this pocket does not contain pure gas but a mixture at thermodynamic equilibrium. For this test case, thermal effects are strong.

The value of dP/dT , evaluated with a thermodynamic table on the temperature interval [250K, 355K], is set to 300 Pa/K . The volume and mass fractions of gas, pressure, temperature, velocity and speed of sound evolutions are plotted in Fig. 3, at time t = 1.5 ms. No differences appear on the void ratio between models whereas the mass fraction simulated by our models is twice higher in comparison with the two-fluid solution. It is due to differences in the evaluation of the gas density. In our models, ρ v is function of the mixture temperature through the stiffened gas EOS while the gas density is function of the gas temperature for the two-fluid model.

Shock-cavitation interaction, | u |= 100 m/s

This case is similar to the previous one, except that the two ends of the tube are simultaneously closed once the flow starts. Therefore, a shock created at each end moves towards the center, resulting in shock-cavitation interaction and cavitation collapse. Both evaporation and condensation processes can be investigated.

The flow is initially quasi pure water (the initial fraction of gas is 10 -10 ) and soon changes phase into a vapour-liquid mixture at the center, and then reverting back into a pure liquid after the cavitation collapse. A similar test case was depicted in [START_REF] Goncalves | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF].

A uniform mesh of 5000 cells is used and the time step is set to 10 -8 s. The parameter c baro is set to 1.31 m/s for the 4-equation barotropic model, as previously.

Volume fraction obtained with both models are plotted at different times in Fig. 5.

As the cavitation pocket grows, up to time t = 0.3 ms, solutions are similar. After time t = 0.3 ms, the shocks created at the ends meet the rarefaction waves generated at the center, and then interacts with the expanding cavitation interface. The cavitation collapse begins. The simulation obtained with the 4-equation barotropic model predicts the decrease of the void ratio. At time t = 0.7 ms, the maximum void ratio value is close to 0.4. On the contrary, using the 4-equation SG model the decrease is badly reproduced. The maximum value at time t = 0.7 ms is 0.9.

The 4-equation SG model is able to produce and to transport the void ratio but it does not destroy the void ratio in the flow where the pressure is greater than the vapour pressure. We propose to modify the formulation adding a destruction term:

ṁ = ρ l ρ v ρ l -ρ v ( 1 - c 2 c 2 wallis ) div V -C des ρ v ρ l α Max (0, P -P vap ) 0.5ρ ref U 2 ref ( 32
)
where C des is a tunable parameter. This new formulation does not modify the mixture speed of sound and the eigenvalues of the inviscid system.

The influence of the constant C des is investigated and different values are tested from 0.1 to 1000. Using values smaller than 100, the effect is weak and is not presented.

The evolution of the volume fraction at different times is presented in Fig. 6 using C des =100 and C des =1000. As expected, higher is the constant C des , more intense is the condensation phenomenon. At time t = 0.7 ms, the maximum void ratio value is 0.75 with C des =100 and only 0.1 with C des =1000.

1D inviscid rarefaction test case with freon R-114

A double rarefaction tube problem is considered with freon R-114 similarly to those given in Table 2. The quantities have been evaluated with a saturation table at the reference temperature.

Velocity | u |= 10 m/s

The value of dP/dT , evaluated with a thermodynamic table on the temperature

Velocity | u |= 100 m/s

The value of dP/dT , evaluated with a thermodynamic table on the temperature interval [50K, 293K], is set to 500 Pa/K. Void ratio, velocity, temperature and pressure evolutions are presented in Fig. 8 at time t =1.5 ms. Both models provide a similar evolution for the void ratio and the velocity profiles. As previously, the temperature depression is more intense using the 4-equation barotropic model. It reaches 260 K with this model and 240 K using the 4-equation SG model. This difference on the cooling effect leads to a more intense pressure drop in the cavitation pocket.

2D viscous turbulent Venturi case

Experimental and numerical conditions

The freon R-114 experimental facility of the CREMHyG is a closed loop operating with a reference pressure, obtained by pressurizing a tank with nitrogen gas.

The cavitation tunnel was designed to simulate cavitating flows developing on the blades of space turbopump inducers. The loop is fitted with a test section having the shape of a two-dimensional Venturi, characterized by a convergence angle of 4.3 • and a divergence angle of 4 • (Fig. 9). The edge forming the throat of the Venturi is used to fix the separation point of the cavitation cavity. This geometry is equipped with three probing holes to take various measurements. Optical probes and micro-thermocouples are used to evaluate the local void ratio and the wall temperature, respectively. The uncertainty on the temperature measurement is about ±0.2K [START_REF] Stutz | Analyse de la structure diphasique et instationnaire de poches de cavitation[END_REF][START_REF] Fruman | Estimation of thermal effects in cavitation of thermosensible liquids[END_REF]. Flow conditions and experimental parameters are given in Table 4.

With these parameters, a cavity length around 80 mm was obtained, with a relatively stable aspect.

The freon R-114 is a thermosensitive fluid, which allows to study the thermodynamic effect in ambient conditions. It provides the same temperature depression ∆T * = 1.22 K in comparison with the liquid hydrogen at T ref = 22K. The thermodynamic properties at saturation for freon R-114 are given in Table 2.

All cavitating simulations are steady computations, which are started from the non cavitating numerical solution. The H-type grid contains 251 nodes in the flow direction and 77 in the orthogonal direction (Fig. 10). Numerical parameters are given in Table 4.

Computations have been performed using both 4-equation models in order to obtain a cavity length close to 80 mm. The value of the inlet cavitation parameter σ inlet was around 0.57 for all computations, close to the experimental value. In the following, the numerical solutions are compared with the previous results obtained with a 3equation model [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF] and the experimental data.

Visualization of the pocket

A visualization of the cavitation pocket is proposed in Figure 11 where the contours of the density gradient modulus are plotted (Schlieren-like visualization). The grey scale is the same for all computations. The cavity length is around 80 mm for all simulations. Using the 3-equation model, the interface of the cavity and the closure part of the cavity are more diffuse in comparison with the 4-equation solutions.

Discrepancies appear also on the thickness of the cavity. The 4-equation barotropic model predicts the thinner cavity.

Void ratio profiles

Figure 12 illustrates the numerical void ratio obtained with the three models in comparison with the measurements. At station 1, all models over-predict the cavity thickness and the maximum value of void ratio. According to analyses presented in [START_REF] Stutz | Analyse de la structure diphasique et instationnaire de poches de cavitation[END_REF], the experimental void ratio is certainly under-estimated at this station. At to the maximum value of the void ratio, the 4-equation barotropic model predicts a higher value in comparison with the two other models. The decrease of the void ratio close to the wall is not captured by models. This behaviour was also observed for cold water simulations [START_REF] Goncalves | Modelling for isothermal cavitation with a fourequation modek[END_REF].

Temperature evolution

The temperature deficit profile at station 1 is plotted in Figure 13 for the three models. The wall temperature is in good agreement with the experimental data using The cooling effect due to the vaporization process is clearly observed for all simulations (negative values) and it is stronger using the 4-equation SG model. A large discrepancy appears between the 3-and 4-equation models downstream the cavity in the recompression area. Using the 3-equation model, the temperature deficit is close to zero downstream the cavitation pocket, that is the temperature goes back to the freestream temperature. On the contrary, we observe a warming effect downstream the cavity using both 4-equations models. Locally, the temperature exceeds the freestream temperature and values reach more than 6 K with the 4-equation SG model.

Such a phenomenon was depicted in [START_REF] Petkovsek | IR measurements of the thermodynamic effects in cavitating flow[END_REF] on a hot water Venturi flow. Using high speed infra-red thermography, authors measured a temperature depression of approximately 0.4 K in the vicinity of the throat and a temperature rise up to 1.4 K was recorded in the region of pressure recuperation. The collapse of bubbles causes the rapid recuperation of the temperature.

The same temperature difference is plotted in Figure 15 in a larger domain for the 4-equation barotropic model. The elevation of temperature is extended inside the divergent up to a distance of 0.4 m that is more than four times the cavity length.

The temperature gradient is illustrated in Figure 16 for the three computations. The grey scale is the same for all computations. We clearly observe differences on the behaviour of models in the closure part of the cavitation pocket. Using the 3-equation model, there is no temperature gradient at the end of the pocket. We just see a small gradient at the interface in the vicinity of the throat. With both 4-equation models, strong gradients are put in evidence in the closure part of the pocket and downstream in the recompression area.

Influence of the destruction term

The destruction term is now activated in the both 4-equation models following relation [START_REF] Goncalves | Modelling for isothermal cavitation with a fourequation modek[END_REF]. This term is driven by a parameter C des . Only one value C des = 100 was tested to observe the condensation process. Inside the cavity at both abscissa x = 0.065m and x = 0.075 m, the effect of the destruction term is clearly noticeable on the void ratio maximum value, which is reduced by a factor 2. Similarly, the warming effect is also reduced (about 1 K).

At the abscissa x = 0.095 m, the void ratio profiles provided by both simulations are quasi superposed. Yet, the temperature deficit profiles mark large discrepancies.

The warming effect is well exhibited when C des = 0 whereas a cooling effect (about 1 K) is predicted when the destruction term is activated. One possible explanation is linked to the transient of the computation. A large pocket for which the length reaches 0.2 m is simulated without the destruction term. In the closure part of the pocket an intense warming effect is predicted. When the cavity narrows, the warming effect remains a long time, which is observed at the abscissa x = 0.095m and x = 0.135 m. For this last location, with C des = 100 the void ratio is null and the temperature is on its freestream value. On the contrary, without the activation of the destruction term the warming effect is intense and reaches almost 8 K.

At the same locations, the void ratio and temperature profiles computed with the 4-equation SG model are plotted in Figure 18. At both abscissa x = 0.065 m and x = 0.075 m, the void ratio profiles provided by simulations are quasi superposed.

The destruction of void ratio is not observed. As a consequence, the temperature deficit profiles are quite similar. At the abscissa x = 0.095 m and x = 0.135 m, the same behaviour observed previously with the 4-equation batrotropic model is put in evidence. With similar void ratio profiles at location x = 0.095 m, temperature evolutions are opposite: a warming effect (about 5 K) without the destruction term and a cooling effect (around 1 K) when the destruction term is used. At the last abscissa, a strong temperature elevation about 8 K is predicted when C des = 0. Using the destruction term, pure liquid at the freestream temperature is computed.

The temperature difference T -T ref is illustrated in Figure 19 for both simulations using the destruction term. We can observe the warming effect located on the closure part of the cavitation pocket. This phenomenon is not extended as previously simulated and the back to the freestream temperature is done on a short distance (less than one cavity length). Yet, as the experimental temperature field is unknown, it is impossible to calibrate the value of the parameter C des .

Conclusion

In -A new phenomenon was simulated with the 4-equations models: a warming effect was exhibited downstream the cavity in the recompression area. The intensity of this warming effect is higher than the cooling effect observed near the Venturi throat. This rise of temperature is due to the collapse of bubbles in the closure part of the pocket and recent infra-red measurements highlighted this phenomenon inside another Venturi geometry in which the running fluid was hot water.

-The distance where this temperature elevation was simulated depends on the condensation modelling. Without the activation of a destruction term, the warming effect was extended more than 4 times the cavity length. Using a constant C des = 100 in the destruction term, the distance became smaller than one cavity length. Unfortunately, we need to know the temperature field to calibrate the destruction parameter.

Finally, the new LEGI models are very attractive to study thermodynamic effects and cryogenic cavitation. Additional works are in progress to pursue comparative analyses between cavitation models and to improve the model calibration. We deduce an expression of the speed of sound:

Appendix

c 2 = ( ∂P ∂ρ ) s = h - ( ∂ρe ∂ρ ) P ( ∂ρe ∂P ) ρ
The derivatives of the void ratio have to be evaluated : ( ∂α ∂ρ

) P = -1 ρ L -ρ V ( ∂α ∂P ) ρ = -1 ρ V -ρ L dT dP [ α dρ V dT + (1 -α) dρ L dT ]
Using the stiffened gas EOS, we have the following equalities:

( ∂ρe ∂P ) ρ = α ( ∂ρ V e V ∂P ) ρ + (1 -α) ( ∂ρ L e L ∂P ) ρ + (ρ V h V -ρ L h L ) ( ∂α ∂P ) ρ = 1 γ -1 + ρ V h V -ρ L h L ρ L -ρ V dT dP [ α dρ V dT + (1 -α) dρ L dT ] ( ∂ρe ∂ρ
) P = ∂ ∂ρ [ α ( P γ V -1 + ρ V q V + γ V γ V -1 P V ∞ ) + (1 -α) ( P γ L -1 + ρ L q L + γ L γ L -1 P L ∞ )] = ρ L h L -ρ V h V ρ L -ρ V 29
Finally, the speed of sound is:

ρc 2 = 1 ( ∂ρe ∂P ) ρ [ ρ V ρ L (ρ L -ρ V ) (h V -h L )
]

-P v ) + θ(T v -T l ) KdivV KdivV analytical α equation + ṁ/ρ I + ṁ/ρ I + ṁ/ρ I α = (ρ-ρ l ) (ρv-ρ l ) ṁ ν(g v -g l ) ( 1 - c 2 c 2 wallis ) divV ( 1 - c 2 c 2 wallis ) divV - metastable liquid - - - states vapour vapour vapour -

4-equation SG

4-equation SG

  temperature of water is T ref = 355 K. A weak volume fraction of vapor α =0.01 is initially added to the liquid. The initial discontinuity is set at 0.5 m, the left velocity and the right velocity have the same value but an opposite sign. Two velocities are tested: u = ±2 m/s and u = ±100 m/s. The vapour pressure P vap (T ref ) is set to 51000 Pa at the initial temperature.

  terval [350K, 355K], is set to 2000 Pa/K. Void ratio, velocity, temperature, pressure, speed of sound and mass fraction of gas evolutions are shown in Fig. 2 at time t =3.2 ms. Results are compared with the two-fluid solution computed by Zein et al. [5].

For the pressure

  profiles, the pressure drop under P vap (T ref ) is around 0.3 bar with both models, in close agreement with the two-fluid solution. Discrepancies appear for the velocity profile in comparison with the two-fluid solution: variations across the evaporation front are stiffer with our models. As previously observed, the SG speed of sound is slightly smaller than the Wallis one near the evaporation fronts. Inside the cavitation pocket, the barotropic speed of sound is close to the Wallis one whereas the SG velocity is twice lower. The temperature drop inside the cavitation pocket due to the phase transition is very large. It reaches more than 70 K with the 4-equation SG model and 90 K with the 4-equation barotropic model. For this case, the difference with the isothermal solution presented in [31] is very strong. Solutions between isothermal and non isothermal 4-equation barotropic models are illustrated in Fig. 4. For the pressure profiles, the pressure drop under P vap (T ref ) is around 0.05 bar with the isothermal model. Moreover the temperature drop is very intense and lead to negative values.

  presented previously in water. The tube is filled with liquid freon at ambient tem-perature T ref = 293 K and the pressure is set to 3 bar. A weak volume fraction of vapor α =0.01 is initially added to the liquid. The initial discontinuity is set at 0.5 m and two velocities are tested: u = ±10 m/s and u = ±100 m/s. The vapour pressure P vap (T ref ) is set to 181000 Pa. The mesh contains 5000 cells. The time step is set to 10 -7 s. The value of c baro is set to 1.66 m/s for the 4-equation barotropic model. The parameters of the stiffened gas EOS and saturation values for densities are

  interval [283K, 293K], is set to 4720 Pa/K. Void ratio, velocity, temperature and pressure evolutions are shown in Fig. 7 at time t =2.4 ms. Both models provide a similar evolution for the void ratio and the velocity profiles. The pressure drop in the cavitation area is more pronounced with the 4-equation barotropic model. It is due to a higher temperature drop. It reaches 9 K with this model instead of 7 K with the 4-equation SG model. Using the B-factor theory and a maximum void ratio value equal to 0.8, the ideal temperature depression is around 4.85 K. Both models tend to overestimate this value.

stations 2 and 3 ,

 3 the cavity thickness is clearly better predicted by the 4-equation barotropic model. Using the 3-equation model, the thickness is over-estimated with a factor 2. An intermediate result is provided by the 4-equation SG model. As regard

the 3 -

 3 equation model and the 4-equation barotropic model. Using the 4-equation SG model, the wall temperature depression is about 4.2K instead of 2.1K for the experimental data. Moreover, the shape of the temperature profile marks discrepancies. The temperature deficit is almost constant in a large part of the cavity from the wall up to y = 0.002 m.

Figure 14 shows

 14 Figure 14 shows the temperature deficit T -T ref inside the divergent of the Venturi for the three computations. The grey scale is the same for all computations.

Figure 17 illustrates

 17 Figure 17 illustrates the void ratio and temperature deficit profiles computed with the 4-equation barotropic model. Profiles are given at different locations in the di-

  this paper, a 4-equation model was developed to study cavitation in thermosensitive fluid. This model is composed by three conservations law for mixture quantities (mass, momentum, total energy) and an additional transport equation for the vapour volume fraction, where mass transfer rate due to cavitation is modelled. The generic model is based on the assumption of proportionality of the mass transfer with the divergence of velocity. Two formulations were proposed using two equations of state: a modified barotropic relation and a mixture of stiffened gas. The vapour pressure is assumed to vary linearly with the mixture temperature through a parameter dP vap /dT . Models have been implemented in a compressible Euler and RANS solvers and have been applied for the simulation of various cavitating problems (inviscid and turbulent cases). First validations on inviscid one-dimensional cases shown the ability of models to simulate the cavitation development in which the running fluid is hot water and freon R-114. Comparisons with two-fluid solutions illustrated the good behaviour of models. Secondly, RANS simulations were performed to study a quasi stable cavitation pocket developing along a Venturi geometry in which the working fluid is freon R-114. Numerical results obtained from the new models have been validated against experimental data and 3-equation computations previously performed in our team. This test-case lead to different concluding remark: -About the cavity thickness, the new models and especially the 4-equation barotropic model clearly improved the prediction in comparison with the 3-equation simulation. -The wall temperature deficit was well simulated by the 3-and 4-equation barotropic models. Yet, the solution obtained with the 4-equation SG model badly reproduced the wall value (over-estimation of a factor 2).

dP

  Appendix A: the speed of sound in a mixture of stiffened gas Starting from the usual thermodynamic relation de = T ds + P ρ 2 dρ or d(ρe) = ρT ds + hdρ And with the differential of ρe:We can obtained the differential of the pressure P :

Figure 2 :

 2 Figure 2: Water-gas double rarefaction with cavitation |u| = 2 m/s, models comparison, mesh 5000 cells, t = 3.2ms. Void ratio, pressure, velocity, temperature, speed of sound and mass fraction of gas.

Figure 3 :

 3 Figure 3: Water-gas double rarefaction with cavitation |u| = 100 m/s, models comparison, mesh 5000 cells, t = 1.5 ms. Void ratio, pressure, velocity, temperature, speed of sound and mass fraction of gas.

Figure 4 :

 4 Figure 4: Water-gas double rarefaction with cavitation |u| = 100 m/s, isothermal versus non isothermal models, mesh 5000 cells, t = 1.5 ms. Pressure and temperature.

Figure 7 :Figure 8 :

 78 Figure 7: Freon R114 double rarefaction with cavitation |u| = 10 m/s, models comparison, mesh 5000 cells, t = 2.4 ms. Void ratio, mixture pressure, temperature and velocity.

Figure 9 :

 9 Figure 9: Schematic view of the Venturi profile.

Figure 11 :Figure 12 :

 1112 Figure 11: Modulus of the density gradient (kg/m 4 ), models comparison, Venturi case.

Figure 14 :Figure 17 :

 1417 Figure 14: Temperature difference T -T ref (K) inside the divergent, models comparison, Venturi case.

Figure 18 :

 18 Figure 18: Void ratio (left) and temperature deficit (right) profiles at different locations, influence of the destruction term, 4-equation SG model, Venturi case.

Figure 19 :

 19 Figure 19: Temperature difference T -T ref (K) inside the divergent, influence of the destruction term, Venturi case.

Table 1 :

 1 Characteristics of models

	models	2-fluid	4-eqt baro	4-eqt SG	3-eqt baro
	solved	2 masses	1 mass	1 mass	1 mass
	equations	1 moment.	1 moment.	1 moment.	1 moment.
		2 energies	1 energy	1 energy	1 energy
		+ α	+α	+α
	mixture EOS SG	baro	SG	baro
	source term	µ(P l

Table 2 :

 2 Parameters of the stiffened gas EOS for freon R114 at T = 293K.

	liquid 1.4 1.21 10 8 -6.901 10 4 984	1469.1
	vapor 1.07 0	1.424 10 5 700	13.52

γ P ∞ (Pa) q (J/kg) C p (J/K.kg) ρ sat (kg/m 3 )

Table 3 :

 3 Parameters of the stiffened gas EOS for water at T = 355K.

	liquid 2.35 10 9	-0.1167 10 7 4267	1149.9
	vapor 1.43 0	0.2030 10 7 1487	0.31

γ P ∞ (Pa) q (J/kg) C p (J/K.kg) ρ sat (kg/m 3 )

Table 4 :

 4 Flow configuration, experimental and numerical parameters, Venturi case

	inlet velocity V inlet

Figure 1: Mixture speed of sound comparison, freon R-114.

eqt baro 4-eqt SG Zein