Quentin Berger
email: quentin.berger@sorbonne-universite.fr

Loïc Béthencourt
email: loic.bethencourt@sorbonne-universite.fr

An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables

Keywords: Randoms walks, exchangeability, fluctuation theory, persistence problem, additive functionals, Markov chains. MSC2020 AMS classification: 60G50, 60J55, 60G09, 60J10

We revisit here a famous result by Sparre Andersen on persistence probabilities P(S k ≥ 0 ∀ 0 ≤ k ≤ n) for symmetric random walks (S n ) n≥0 . We give a short proof of this result when considering sums of random variables that are only assumed exchangeable and sign-invariant. We then apply this result to the study of persistence probabilities of (symmetric) additive functionals of Markov chains, which can be seen as a natural generalization of integrated random walks.

Introduction

We discuss in this paper several results on persistence problems for symmetric random walks, in particular the well known (but nonetheless striking) fluctuation theorem due to Sparre Andersen [START_REF] Andersen | On the fluctuations of sums of random variables II[END_REF]. To introduce the result, let ξ 1 , . . . , ξ n be real random variables and let us denote S 0 = 0, and S k = k i=1 ξ i for 1 ≤ k ≤ n. Then, one is interested in estimating the persistence probabilities P S k > 0 for all 1 ≤ k ≤ n or P S k ≥ 0 for all 1 ≤ k ≤ n . Sparre Andersen's result [START_REF] Andersen | On the fluctuations of sums of random variables II[END_REF] states that if the (ξ i ) 1≤i≤n are i.i.d., symmetric and have no atom, then these probabilities do not depend on the law of ξ.

Theorem A. If the (ξ i ) 1≤i≤n are i.i.d. and if the distribution of ξ i is symmetric and has no atom, then we have

P S k > 0 for all 1 ≤ k ≤ n = (2n -1)!! (2n)!! = 1 4 n 2n n .
In fact, [START_REF] Andersen | On the fluctuations of sums of random variables II[END_REF] proves a number of results for exchangeable random variables, but the independence plays an important role in obtaining fluctuation's results, such as arcsine laws or persistence probabilities. Chapter XII in Feller's book [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF] provides a streamlined approach to Sparre Andersen's result, using a duality argument together with a (purely combinatorial) cyclic lemma, see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]1 XII.6]: these two ingredients only require the law of (X 1 , . . . , X n ) to be exchangeable. Independence is required in the last step of the proof, in order to obtain that the ladder epochs defined iteratively by T + 0 = 0 and T + k = min{n ≥ T + k-1 , S n > max 0≤j<n {S j }} form a renewal sequence. The following is then deduced, see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]XII.7 Theorem B can be seen as a particular case of a Wiener-Hopf factorization, also known as Spitzer-Baxter formula, which gives the joint Laplace transform/characteristic function of the first ladder epoch and ladder height, see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]XII.9]. Note that if the law of X i is symmetric and has no atom, one has P(S n > 0) = 1 2 for all n, so the generating function of T + 1 is given by E[s T + 1 ] = 1 -√ 1s. Let us mention that another proof of Theorem A is presented in [START_REF] Dembo | Persistence of iterated partial sums[END_REF]Prop. 1.3]: the proof is remarkably simple and elegant and relies on a decomposition with respect to the first time that (S k ) k≥0 hits its minimum min 1≤j≤n {S j }, using the independence of the (ξ i ) 1≤i≤n . The goal of our paper is to present a version of Theorem A valid for exchangeable and sign-invariant random variables (ξ i ) 1≤i≤n , going beyond the independent and symmetric setting. Sparre Andersen was already aware of this result, see [START_REF] Andersen | On the fluctuations of sums of random variables II[END_REF]Thm. 4] (its proof is however a bit laconic), but we give here a short and self-contained proof (with no combinatorial lemma), taking inspiration from [START_REF] Dembo | Persistence of iterated partial sums[END_REF]. We then present some application to persistence probabilities for symmetric additive functionals of birth-death chains, giving in particular a simple proof of Sinai's result [START_REF] Sinai | Distribution of some functionals of the integral of a random walk[END_REF] on the integrated simple random walk.

The case of exchangeable and sign-invariant random variables

Below, we consider examples in which there is no independence: one is however still able to obtain the same conclusion as in Theorem A, using some weaker but natural assumption. Let us introduce some terminology: the law P of (ξ 1 , . . . , ξ n ) is said to be: (E) exchangeable if for any permutation σ ∈ S n , (ξ σ(1) , . . . , ξ σ(n) ) has the same distribution as (ξ 1 , . . . , ξ n );

(S) sign-invariant if for any ε = (ε 1 , . . . , ε n ) ∈ {-1, 1} n , (ε 1 ξ 1 , . . . , ε n ξ n ) has the same distribution as (ξ 1 , . . . , ξ n ).
One can think about these two assumptions as an invariance of P under the action of the groups: of permutations S n for condition (E); of sign changes {-1, 1} n for condition (S).

The following result is actually already mentioned in [START_REF] Andersen | On the fluctuations of sums of random variables II[END_REF]Thm. 4]: the present paper is meant as a way to put more emphasis on this result; we then provide some applications.

Theorem 2.1. Assume that (ξ 1 , . . . , ξ n ) satisfies conditions (E)-(S). Then, if P has no atom,

P S k > 0 for all 1 ≤ k ≤ n = (2n -1)!! (2n)!! = 1 4 n 2n n .
In general, if P has atoms, we have

P S k > 0 for all 1 ≤ k ≤ n ≤ (2n -1)!! (2n)!! = 1 4 n 2n n ≤ P S k ≥ 0 for all 1 ≤ k ≤ n .
One can easily see that both conditions (E)-(S) are necessary to obtain the statement of Theorem 2.1, see for instance Section 4.2 below.

Remark 2.2 (About sign-invariance). The sign-invariance of (ξ 1 , . . . , ξ n ) can be seen as a strong form of symmetry for the law of (ξ 1 , . . . , ξ n ). Let us mention the article [START_REF] Berman | Sign-invariant random variables and stochastic processes with sign-invariant increments[END_REF] which derives some properties of sign-invariant sequences: in particular, [6, Lem. 1.2] tells that sign-invariant (ξ 1 , . . . , ξ n ) are independent conditionally on (|ξ 1 |, . . . , |ξ n |). In other words, any exchangeable and signinvariant law can be obtained by taking a random vector (Z 1 , . . . , Z n ) (that can be constrained to have

0 ≤ Z 1 ≤ • • • ≤ Z n )
and by shuffling and changing the signs of its coordinates randomly, that is setting

ξ i = ε i Z σ(i) for 1 ≤ i ≤ n,
= ±1) = 1 2
) and σ is a random uniform permutation of {1, . . . , n} (i.e. such that

P(σ = ν) = 1
n! for all ν ∈ S n ). Remark 2.3 (Finite vs. infinite sequences). The law of an infinite sequence (ξ i ) i≥1 of random variables is called exchangeable, resp. sign-invariant, if for any n ≥ 1 the law of (ξ 1 , . . . , ξ n ) is exchangeable, resp. sign-invariant. By de Finetti's theorem, one knows that the law of ξ = (ξ i ) i≥1 is exchangeable if and only if it is conditionally i.i.d.: in other words, there exists a random probability distribution µ such that P(ξ ∈ • | µ) = µ ⊗∞ , where µ ⊗∞ denotes the law of an infinite sequence of i.i.d. random variables with law µ. In [5, Thm. 1], it is shown that an exchangeable sequence (ξ i ) i≥1 is sign-invariant if and only if µ is almost surely symmetric. From this, it is easy to derive Theorem 2.1 for an exchangeable and sign-invariant sequence (ξ i ) i≥1 , simply by working conditionally on the realization of µ. Hence, the truly remarkable fact about Theorem 2.1 is that it holds for a finite exchangeable and sign-invariant (ξ 1 , . . . , ξ n ).

Persistence probabilities of f -integrated birth-death chains

Let us consider (X n ) n≥0 a birth and death Markov chain, starting from X 0 = 0: it is a Markov chain on Z such that |X n -X n-1 | ≤ 1, with transition probabilities p x = p(x, x + 1), q x = p(x, x -1), r x = p(x, x) = 1p xq x for x ≥ 1.

We assume that (X n ) n≥0 is symmetric, in the sense that (-X n ) n≥0 has the same distribution as (X n ) n≥0 ; in other words, the transition probabilities verify p(x, y) = p(-x, -y) for any x, y ∈ Z. For x = 0, we also set p 0 = p(0, 1) ∈ (0, 1 2 ], q 0 = p(0, -1) = p 0 and r 0 = p(0, 0) = 1 -2p 0 . Let f : Z → R be any anti-symmetric function which preserves the sign of x, i.e. such that f (x) > 0 for x > 0 and f (x) < 0 for x < 0 (and naturally f (0) = 0). Then we define the f -integrated Markov chain, or additive functional, as follows: ζ 0 = 0 and, for n ≥ 1,

ζ n := n i=1 f (X i ) . (3.1)
We are now interested in the persistence (or survival) probabilities

P(ζ k > 0 for all 1 ≤ k ≤ n) or P(ζ k ≥ 0 for all 1 ≤ k ≤ n) .
A classical, well-studied example is when (X n ) n≥0 is the simple symmetric random walk and f is the identity: then (ζ n ) n≥0 is the integrated random walk and the persistence probabilities are known to be of order n -1/4 , see [START_REF] Sinai | Distribution of some functionals of the integral of a random walk[END_REF]. We prove the following result.

Theorem 3.1. Let (X n ) n≥0 be a symmetric recurrent birth-death chain and let f be an antisymmetric function which preserves the sign of x. Let p := P(X 1 ≥ 0) = 1p 0 and let g(m) :=

P ζ k > 0 for all 1 ≤ k ≤ n ≤ E g(L n ) ≤ p -1 P ζ k ≥ 0 for all 1 ≤ k ≤ n ,
where

L n := n k=1
1 {X k =0} is the local time of the chain at 0 up to time n. We also have

P ζ k > 0 for all 1 ≤ k ≤ n ≥ p(1 -p)E g(L n -1) 1 {Ln≥1} .
Let us mention that our approach is based on an excursion decomposition of the process (X n ) n≥0 .

Our results would still be valid for Markov chains that "cannot jump above 0", i.e. such that

P(X n ≤ 0 | X n-1 = x) = P(X n = 0 | X n-1 = x) for any x ∈ N.
We have chosen to stick with the birth-death setting since it is the most natural example of such chains (and already contains a wide class of behaviors).

In the Theorem 3.1 above, since we have

g(m) = 1 4 m 2m m ∼ (πm) -1/2
as m → ∞ and since the chain is assumed to be recurrent (which implies that

L n → ∞ a.s.), it is tempting to replace the term E[g(L n )] by (1 + o(1))π -1/2 E[(1 + L n ) -1/2
]. However, one needs a bit of caution: let us give a sufficient condition (often verified in practice) to obtain such a result1 . Lemma 3.2. Assume that:

(i) there is a sequence

(b n ) n≥1 , lim n→∞ b n = +∞, such that ( 1 bn L n ) n≥1 converges in distribution to a random variable X ; (ii) for all t > 0 we have sup n≥1 P( 1 bn (1 + L n ) ≤ t) ≤ Ch(t)
, for some constant C > 0 and some non-decreasing function h that verifies

∞ 0 h(u)u -3/2 du < +∞. Then the random variable X verifies E[X -1/2 ] < +∞ and, as n → ∞, E g(L n -1) 1 {Ln≥1} ∼ E g(L n ) ∼ 1 √ π E (1 + L n ) -1/2 ∼ 1 √ πb n E X -1/2 . If (X n ) n≥0 is positive recurrent, then E[g(L n )] ∼ √ E[τ 1 ]
√ πn , where τ 1 := inf{n ≥ 1, X n = 0}.

Example 3.3. We give below a class of example where Lemma 3.2 can be applied, but let us comment on the case of the simple symmetric random walk (X n ) n≥0 . In that case, Lemma 3.2 is verified with b n = n/2 and X = |Z|, Z ∼ N (0, 1); and with the function h(u) = min(u, 1). We then get that the persistence probabilities for the integrated random walk verify

(1 + o(1)) 1 4 c 0 n -1/4 ≤ P ζ k > 0 for all 1 ≤ k ≤ n ≤ (1 + o(1))c 0 n -1/4 , (3.2
)

with c 0 := π -1/2 2 1/4 E[|Z| -1/2 ] = π -1 Γ(1/4
). This recovers a result by Sinai [START_REF] Sinai | Distribution of some functionals of the integral of a random walk[END_REF] (in the case f (x) = x) and makes the constants explicit.

Remark 3.4. We stress here that the asymptotic bounds that we obtain combining Theorem 3.

A class of examples: Bessel-like random walks

Let us give more precise result in the case of symmetric Bessel-like random walk, see [START_REF] Alexander | Excursions and local limit theorems for Bessel-like random walks[END_REF][START_REF] Hodges | Recurrence-time moments in random walks[END_REF][START_REF] Lamperti | A new class of probability limit theorems[END_REF]. This is a class of birth-death chains that includes the simple symmetric random walk. They have the following transition probabilities: for x ≥ 1

p x := p(x, x + 1) = 1 2 1 - δ + ε x 2x , p(x, x -1) = 1 -p x , (3.3)
where δ ∈ R and ε x is such that lim x→∞ ε x = 0; we take p(-x, -x -1) = p(x, x + 1) and p(0, 1) = p(0, -1) = 1 2 , for symmetry reasons. We also assume uniform ellipticity, i.e. there is some η > 0 such that p x ∈ [η, 1η] for all x ∈ Z. The parameter δ is called the drift parameter and we have the following behavior: the walk (

X n ) n≥0 is transient if δ < -1, recurrent if δ > -1, positive recurrent if δ > 1;
in the cases δ = -1 and δ = 1, the behavior depends on the function ε x .

More precisely, letting

λ x = x k=1 px 1-px , the random walk (X n ) n≥1 is recurrent if and only if ∞ x=1 λ x = +∞.
Moreover, with the notation (3.3), there is a constant K 0 > 0 such that

λ x = x k=1 1 -p k p k ∼ K 0 x δ ℓ(x) -1 as x → ∞, where ℓ(x) := exp x k=1 ε k k . (3.4)
Note that ℓ is a slowly varying function. Then, in [START_REF] Alexander | Excursions and local limit theorems for Bessel-like random walks[END_REF], the sharp tail of the distribution of τ 1 := inf{n ≥ 1, X n = 0} is derived: we reported the result in Section 5.3 below, see (5.1). This enables us to verify that L n satisfies the assumptions of Lemma 3.2 and to show the following result. We focus on the null-recurrent case, since the positive recurrent case is treated in Lemma 3.2. Proposition 3.5. Assume that (X n ) n≥0 is null-recurrent (necessarily δ ∈ [-1, 1]). Then let us set κ := 1+δ 2 ≥ 0 and define

b n :=        nµ(n) -1 if δ = 1 , K 0 Γ(κ) 2 1-κ Γ(1-κ) n κ ℓ( √ n) -1 if δ ∈ (-1, 1) , K -1 0 ν(n) -1 if δ = -1 , with µ(n) := n k=1 P(τ 1 ≥ k) , ν(n) := x≤n, x even 1 xℓ( √ x) .
(3.5)

In the case δ = 1, µ(n) is slowly varying (and lim n→∞ µ(n) = E[τ 1 ]); in the case δ = -1, ν(n) is
slowly varying and the recurrence is equivalent to having lim n→∞ ν(n) = +∞.

We then have that

(1 + o(1)) 1 4 c κ (b n ) -1/2 ≤ P ζ k > 0 for all 1 ≤ k ≤ n ≤ (1 + o(1)) c κ (b n ) -1/2 , with: c κ = 1 if δ = 1; c κ = E[Z κ/2 ] if δ ∈ (-1, 1), where Z is a one-sided κ-stable random variable of Laplace transform e -t κ ; c κ = √ π if δ = -1.

Further comments and comparison with the literature

Let us stress that there are several directions in which one could extend our results. First, one could consider more general underlying random walks or Markov chains, for instance random walks with non-necessarily symmetric increments or Markov chains that can "jump over 0". One could also consider more general functions f , not necessarily symmetric. Another room for improvement is to obtain sharp asymptotics for the persistence probabilities, i.e. for instance finding the correct constant c κ such that

P(ζ k > 0 for all 1 ≤ k ≤ n) ∼ c κ b -1/2 n in Proposition 3.5.
A big part of the literature has considered the case of integrated random walks, that is considering the Markov chain X n = n k=1 ξ k with (ξ k ) k≥0 i.i.d. random variables and ζ n = n i=1 X i (i.e. taking f (x) = x), starting with the work of Sinai [START_REF] Sinai | Distribution of some functionals of the integral of a random walk[END_REF]. Under the condition that the ξ k 's are centered with a finite second moment, the persistence probability P(ζ k > 0 for all 1 ≤ k ≤ n) has been proven to be of order n -1/4 in [START_REF] Dembo | Persistence of iterated partial sums[END_REF], and the sharp asymptotic ∼ c 1 n -1/4 in [START_REF] Denisov | Exit times for integrated random walks[END_REF]. The case where ξ k does not have a finite second moment remains mostly open, except in some specific (one-sided) cases, see e.g. [START_REF] Dembo | Persistence of iterated partial sums[END_REF][START_REF] Vysotsky | Positivity of integrated random walks[END_REF]. Our result can be seen as an extension to persistence problems for additive functionals of Markov chains (X n ) n≥0 ; with the major restriction of symmetry and of the fact that the underlying Markov chain cannot jump above 0 (but with the advantage of having an elementary proof).

Another line of works considered persistence problems for additive functionals of continuoustime Markov processes (X t ) t≥0 . Let us mention [START_REF] Goldman | On the first passage of the integrated wiener process[END_REF] that considered the case of an integrated Brownian motion, and [START_REF] Isozaki | Asymptotic estimates for the first hitting time of fluctuating additive functionals of brownian motion[END_REF][START_REF] Profeta | Persistence and exit times for some additive functionals of skew bessel processes[END_REF] who considered the f -integral of a Brownian motion or a skew-Bessel process respectively, for the (possibly asymmetric) functional f (x) = |x| γ (c + 1 {x>0}c -1 {x<0} ) for some γ > -1. More recently, in [START_REF] Berger | Persistence problems for additive functionals of onedimensional Markov processes[END_REF], we have pushed further the existing techniques (based on a Wiener-Hopf decomposition of a bi-variate Lévy process associated to the problem): we obtained the sharp asymptotics for persistence probabilities for a wide class of Markov processes (including one-dimensional generalized diffusions, see [START_REF] Itô | Diffusion processes and their sample paths: Reprint of the 1974 edition[END_REF]) and of functions f . In particular, [4, Example 6] shows that the result applies to continuous-time birth-death processes, giving the existence of the constant c κ mentioned above. The present article can be seen as a elementary approach to obtaining a sub-optimal result.

4 Exchangeable and symmetric sequences: proof of Theorem 2.1

Let us introduce some notation. For n ∈ N and for a fixed x = (x 1 , . . . , x n ) ∈ R n , we define an exchangeable and symmetric vector ξ = (ξ 1 , . . . , ξ n ) with law denoted P (x) by permuting the coordinates of (x 1 , . . . , x n ) by a random uniform permutation and by changing the signs of the coordinates uniformly at random. More precisely, let (ε i ) 1≤i≤n be i.i.d. random variables with law P(ε 1 = 1) = P(ε i = -1) = 1 2 and let σ be a random permutation of {1, . . . , n} with uniform distribution P(σ = ν) = 1 n! for all ν ∈ S n , independent of (ε i ) 1≤i≤n : we then define

(ξ 1 , . . . , ξ n ) := (ε 1 x σ(1) , . . . , ε n x σ(n) ) . (4.1)
Note that we can restrict to the case where x i ∈ R + for all 1 ≤ i ≤ n.

We then construct the random walk S k = k i=1 ξ i for any 0 ≤ k ≤ n, and we are interested in the persitence probabilities p n (x) := P (x) S k > 0 for all 1 ≤ k ≤ n and pn (x) := P (x) S k ≥ 0 for all 1 ≤ k ≤ n .

We set by convention these probabilities equal to 1 for n = 0.

Proposition 4.1. If x = (x 1 , . . . , x n ) is such that i∈I x i = j∈J
x j for all I, J ⊂ {1, . . . , n} with I = J (H) (by convention i∈∅ x i = 0), then we have that

p n (x) = pn (x) = 1 4 n 2n n = (2n -1)!! (2n)!! = n i=1 1 - 1 2i
.

and in particular it does not depend on x. In general, we have p n (x) ≤ (2n-1)!! (2n)!! ≤ pn (x). Theorem 2.1 follows directly from Proposition 4.1, noting that the law P of (ξ 1 , . . . , ξ n ), conditionally on |ξ| = (|ξ 1 |, . . . , |ξ n |), is P (|ξ|) .

Proof of Proposition 4.1

Our proof is greatly inspired by that of [START_REF] Dembo | Persistence of iterated partial sums[END_REF]Prop. 1.3]. We start with the first statement: we are going to prove by recurrence on n that for any x = (x 1 , . . . , x n ) that verifies the assumption (H), the quantities p n (x) = pn (x) do not depend on x. The statement is trivial for n = 1 since we have P (x) (ξ 1 > 0) = P (x) (ξ 1 ≥ 0) = 1 2 if x = 0, so we directly proceed to the induction step. Let us fix n ≥ 2 and some x = (x 1 , . . . , x n ) that verifies assumption (H). We now apply a path decomposition used in [START_REF] Dembo | Persistence of iterated partial sums[END_REF]. Let W = min{k, S k = max 1≤i≤n S i }: then, for any ℓ ∈ {0, . . . , n}, we have

{W = ℓ} = {ξ ℓ > 0,ξ ℓ + ξ ℓ-1 > 0, . . . , ξ ℓ + • • • + ξ 1 > 0} ∩ {ξ ℓ+1 ≤ 0, ξ ℓ+1 + ξ ℓ+2 ≤ 0, . . . , ξ ℓ+1 + • • • + ξ n ≤ 0} .
Now, we have no independence at hand, but we can further decompose over permutations of (x 1 , . . . , x n ) with a fixed image I = σ({1, . . . , ℓ}): we get

P (x) (W = ℓ) = I⊂{1,...,n},|I|=ℓ ν 1 :{1,...,ℓ}֒→ →I ν 2 :{ℓ+1,...,n}֒→ →I c 1 n! P j i=0 ε ℓ-i x ν 1 (ℓ-i) > 0 for all 0 ≤ j ≤ ℓ -1 , j i=1 ε ℓ+i x ν 2 (ℓ+i) ≤ 0 for all 1 ≤ j ≤ n -ℓ ,
where ν : A ֒→ → B means that ν is a bijection from A to B. By independence and symmetry of the (ε i ) 1≤i≤n , then recombining the sums over the permutations and using the exchangeability, we get that P (x) (W = ℓ) is equal to

I⊂{1,...,n},|I|=ℓ ℓ!(n -ℓ)! n! P (x I ) S k > 0 for all 1 ≤ k ≤ ℓ P (x I c ) S k ≥ 0 for all 1 ≤ k ≤ n -ℓ ,
where we have defined x J = (x i ) i∈J for any J ⊂ {1, . . . , n}. Since both x I and x I c verify the assumption (H), we can apply the induction hypothesis: we have that p ℓ (x I ), p n-ℓ (x I c ) does not depend on x I , x I c , for 1 ≤ ℓ ≤ n -1. Hence, denoting p ℓ := p ℓ (x I ), p n-ℓ := p n-ℓ (x I c ), we end up with P (x) (W = ℓ) = p ℓ p n-ℓ , for any ℓ ∈ {1, . . . , n -1}. Since we have P (x) (W = n) = P (x) (ξ n > 0, ξ n + ξ n-1 > 0, . . . , ξ 1 > 0) = p n (x) by exchangeability and P (x) (W = 0) = P (x) (S k ≤ 0 for all 1 ≤ k ≤ n) = pn (x) by symmetry, we get

1 = n ℓ=0 P (x) (W = ℓ) = p n (x) + n-1 ℓ=1 p ℓ p n-ℓ + pn (x) .
Using that i∈I x i = j∈J x j for all I, J ⊂ {1, . . . , n} with I = J, we obtain that p n (x) = pn (x): the above identity shows that p n (x) = pn (x) =: p n does not depend on x.

We can now determine the value of p n , as done in [START_REF] Dembo | Persistence of iterated partial sums[END_REF]. From the above, (p n ) n≥0 satisfies the recursive relation 1 = n ℓ=0 p ℓ p n-ℓ for every n ≥ 0 (recall that p 0 = 1). Constructing the generating function, we get that for any |x| < 1,

∞ n=0 x n = 1 1 -x = ∞ ℓ=0 p ℓ x ℓ 2 .
Therefore, the generating function of (p n ) n≥0 is equal to (1x) -1/2 , from which one deduces that

p n = 1 4 n 2n n = (2n-1)!! 2n!!
for all n ≥ 0.

For the general bounds, for any fixed x, for any δ > 0 fixed, one can choose y = y(x, δ) such that x + y verifies the assumption (H) and n i=1 |y i | ≤ δ (take e.g. a typical realization of i.i.d. random variables uniform in [0, δ/n]). Then, we clearly have that

P (x+y) S k ≥ 0 for all 1 ≤ k ≤ n ≥ P (x) S k ≥ δ for all 1 ≤ k ≤ n , ≤ P (x) S k ≥ -δ for all 1 ≤ k ≤ n .
Since x + y verifies assumption (H), we get that the probability on the left-hand side does not depend on x, y (and is equal to (2n-1)!! 2n!! ), so

P (x) S k ≥ δ for all 1 ≤ k ≤ n ≤ (2n -1)!! 2n!! ≤ P (x) S k ≥ -δ for all 1 ≤ k ≤ n .
Since δ is arbitrary, letting δ ↓ 0 concludes the proof.

On the necessity of conditions (E)-(S)

Let us stress that in the construction (4.1) of the exchangeable and symmetric vector (ξ 1 , . . . , ξ n ), the two assumptions are essential:

(a) The signs (ε i ) 1≤i≤n need to be independent. As a counter-example, take

(ε i ) 1≤i≤n uniform on {ω ∈ {-1, 1} n , n i=1 ω i ∈ {2-n, n -2}}, so that ξ is still symmetric. Let x = (x 1 , . . . , x n ) ∈ R n + with x 1 ≥ x 2 ≥ • • • ≥ x n (
this is no restriction by definition of ξ). Then to have S k > 0 for all 1 ≤ k ≤ n, we need to have ε 1 = +1 (which happens with probability 1/2n) and then, since all other signs are ε i = -1, we need to place x 1 in the first position (which happens with probability 1/n): we get

p n (x) = P (x) S k > 0 for all 1 ≤ k ≤ n = 1 2n 2 if x 1 > n i=2 x i , 0 if x 1 ≤ n i=2 x i . Therefore p n (x) depends on x.
(b) The signs (ε i ) 1≤i≤n need to be independent from the permutation σ. As a counter-example, take n = 3 and x = (x 1 , x 2 , x 3 ) ∈ (R + ) 3 with x 1 > x 2 > x 3 > 0: then the probability P (x) (S 1 > 0, S 2 > 0, S 3 > 0) is equal to (one needs to have ε 1 = +1)

P(ε 1 = ε 2 = ε 3 = 1) + P(ε 1 = ε 2 = 1, ε 3 = -1, x σ(3) < x σ(1) + x σ(2) ) + P(ε 1 = ε 3 = 1, ε 2 = -1, x σ(2) < x σ(1) ) + P(ε 1 = 1, ε 2 = ε 3 = -1, x σ(2) + x σ(3) < x σ(1) ) . Now, if the joint distribution of (ε, σ) is such that ε = (ε i ) 1≤i≤3 is uniform on {-1, 1} 3 and P(σ(3) = 3 | ε 1 = ε 2 = +1, ε 3 = -1) = 1, P(σ(2) = 3 | ε 1 = ε 2 = +1, ε 3 = -1) = 1 and P(σ(1) = 1 | ε 1 = +1, ε 2 = ε 3 = -1) = 1, since x 1 > x 2 > x 3 > 0, we get that P (x) (S 1 > 0, S 2 > 0, S 3 > 0) = 1 4 + 1 8 1 {x 1 >x 2 +x 3 } ,
which depends on x. It simply remains to see that the above conditions on the joint distribution of (ε, σ) can be satisfied, which can be checked by hand.

5 Integrated birth and death chains 5.1 Persistence for integrated birth-death chains: Proof of Theorem 3.1

Let us define iteratively τ 0 = 0 and, for k ≥ 1, τ k = min{n > τ k-1 , X n = 0}. Then, we define for k ≥ 1, the random variable

ξ k = τ k i=τ k-1 +1 f (X i ) ,
i.e. the contribution of the k-th excursion of (X n ) n≥0 to the f -integrated Markov chain. Note that by the Markov property, the (ξ k ) k≥1 are i.i.d. We can therefore write

ζ n := n i=1 f (X i ) as ζ n = Ln k=1 ξ k + W n , W n = n i=τ Ln +1 f (X i )
where we recall that L n = n i=1 1 {X i =0} is the local time at 0. Since there is no change of sign during an excursion (recall that (X n ) n≥0 is a birth-death chain), we have that (ζ) n≥0 is monotonous on each interval (τ k-1 , τ k ].

We therefore get the following upper bound: removing the positivity condition on the last segment (τ Ln , n], we get

P(ζ k > 0 for all 1 ≤ k ≤ n) ≤ P ℓ k=1 ξ k > 0 for all 1 ≤ ℓ ≤ L n .
The difficulty here is that the number of terms in the sum is random and that ξ i and L n are not independent. If we condition on L n and on τ Ln , we get that

P ℓ k=1 ξ k > 0 for all 0 ≤ ℓ ≤ L n = E P ℓ k=1 ξ k > 0 for all 0 ≤ ℓ ≤ L n L n , τ Ln .
Then, for any 0 ≤ m ≤ j ≤ n, conditionally on {L n = m, τ m = j}, the random variables (ξ i ) 1≤i≤m are not independent, but they are easily seen to be exchangeable and sign-invariant (thanks to the Markov property and the fact that the chain (X n ) n≥0 is symmetric). We can therefore apply Theorem 2.1 to obtain that

P ℓ k=1 ξ k > 0 for all 0 ≤ ℓ ≤ m L n = m, τ m = j ≤ 1 4 m 2m m =: g(m) .
This enables us to bound the conditional probability and we end up with

P ζ k > 0 for all 1 ≤ k ≤ n ≤ E g(L n ) .
As far as the lower bound is concerned, imposing that the excursion straddling n is non-negative (with probability p) so that W n ≥ 0, and using that the sign of W n is independent from the past by symmetry, we get

P(ζ k ≥ 0 for all 1 ≤ k ≤ n) ≥ pP ℓ k=1 ξ k ≥ 0 for all 1 ≤ ℓ ≤ L n .
As above, conditioning with respect to L n , τ Ln and using Theorem 2.1, we get that

P(ζ k ≥ 0 for all 1 ≤ k ≤ n) ≥ pE g(L n ) .
Let us now derive the last lower bound in Theorem 3.1. On the event {L n ≥ 1}, let us stress that ξ 1 > 0 if and only if X 1 > 0. Then, imposing on this event the first excursion to be positive and the last to be non-negative, we get that

P(ζ k > 0 for all 1 ≤ k ≤ n) ≥ p(1 -p)P ℓ k=2 ξ k > 0 for all 1 ≤ ℓ ≤ L n , L n ≥ 1 .
Then, conditioning with respect to {L n = m, τ m = j}, and using the exchangeability and signinvariance of (ξ i ) 2≤i≤m , we get thanks to Theorem 2.1

P ℓ k=2 ξ k ≥ 0 for all 1 ≤ ℓ ≤ L n L n ≥ 1 L n = m, τ m = j ≥ g(m -1)1 {m≥1} ,
since there are only m -1 terms in the sum. This concludes the proof.

Proof of Lemma 3.2

Let us start by giving the asymptotic of E[(1 + L n ) -1/2 ]. First, we have that

E (1 + L n ) -1/2 = ∞ 0 P (1 + L n ) -1/2 ≥ t dt = ∞ 0 P 1 + L n ≤ t -2 dt = b -1/2 n ∞ 0 P b -1 n (1 + L n ) ≤ u 1 2 u -3/2 du . Now, using that P(b -1 n (1 + L n ) ≤ u) ≤ Ch(u) with ∞ 0 h(u)u -3/2
du < +∞, together with the fact that 1 bn (1 + L n ) converges in distribution to X , we get by dominated convergence that the last integral converges to

1 2 ∞ 0 P(X ≤ u)u -3/2 du = ∞ 0 P(X ≤ t -2 )dt = E X -1/2 ,
where the integral is convergent since

P(X ≤ u) ≤ Ch(u) with ∞ 0 h(u)u -3/2 du < +∞. This proves that E[(1 + L n ) -1/2 ] = b -1/2 n E[X -1/2 ] as n → ∞.
It remains to see that the first expectations are asymptotically equivalent. We simply need to show that the contribution to E[g(L n )] is mostly concentrated on large values of L n . Let us fix K > 0 arbitrarily large, and notice that since g(m) ≤ 1,

E g(L n )1 {Ln≤K} ≤ P(L n ≤ K) = P b -1 n (1 + L n ) ≤ b -1 n (1 + K) ≤ Ch (K + 1)b -1 n .
Now, since h is non-decreasing with 1 0 h(u)u -3/2 du < +∞, we get that h(x)x -1/2 goes to 0 as x ↓ 0. Hence, for any fixed K > 0, we have that

P(L n ≤ K) = o(b -1/2 n
). Choosing K = K n that goes to infinity slowly enough, this proves that

E g(L n ) = E g(L n )1 {Ln>Kn} + o(b -1/2 n ) ,
and similarly for E[(1 + L n ) -1/2 ]. Now, we clearly have that g(m) = (1 + o(1))π -1/2 m -1/2 with the o(1) uniform in m ≥ K n . We therefore end up with

E g(L n ) ∼ π -1/2 E[(1 + L n ) -1/2 ] .
The expectation E[g(L n -1) 1 {Ln≥1} ] is treated similarly.

We conclude by showing that the assumptions are verified in the case where (X n ) n≥1 is positive recurrent. The ergodic theorem already shows that lim n→∞

1 n L n = 1 E[τ 1 ]
a.s., so we can take b n = n. We now show that

P 1 n (1 + L n ) ≤ t = P(1 + L n ≤ tn) ≤ Ch(t) with h(t) = min(1, t 3/4 ).
Let us consider the successive return times τ k = min{n > τ k-1 , X n = 0}, which form a renewal process τ = (τ i ) i≥0 . Then, our starting point is to write that, for any t > 0, we have

P(1 + L n ≤ tn) = P(τ tn > n)
(we omit the integer part in ⌊tn⌋ for simplicity). If t > , then we use a standard bound: let δ(t) > 0 to be chosen in a moment, and write

P(τ tn > n) ≤ P(∃ k ≤ tn, τ k -τ k-1 > δ(t)n) + P tn k=1 (τ k -τ k-1 )1 {τ k -τ k-1 ≤δ(t)n} > n ≤ tnP(τ 1 > δ(t)n) + e -un E
(1 + L n ≤ tn) ≤ C( t δ(t) + e -1 2δ(t) ) for t ≤ 1 4E[τ 1 ]
. Thich concludes the proof, taking δ(t) = t 1/4 .

About Bessel-like random walks: proof of Proposition 3.5

The proof of Proposition 3.5 boils down to the proof of the following lemma, together with an application of Theorem 3.1 and Lemma 3.2. Recall the notation from Section 3, in particular (3.3) and (3.4). Recall also the definition (3.5) of the sequence (b n ) n≥1 .

Lemma 5.1. In the null-recurrent case, ( 1 bn L n ) n≥1 converges in distribution to:

(i) X = 1 if δ = 1; (ii) X = Z -κ if δ ∈ (-1, 1); (iii) X ∼ Exp(1) if δ = -1.
Additionally we have, uniformly in n, P(

:= 1+δ 2 , then as n → ∞ if δ > -1 (κ > 0), P(τ 1 ≥ x) ∼ 2 1-κ K 0 Γ(κ) x -κ ℓ( √ x) , if δ = -1 (κ = 0), P(τ 1 ≥ x) ∼ 1 K 0 ν(x) -1 , (5.1)
where ν(x) is defined in (3.5); also, (X n ) n≥0 is recurrent if and only if lim x→∞ ν(x) = ∞.

Since the tail distribution P(τ 1 ≥ n) is regularly varying with exponent -κ, we have the following convergences in distribution for (τ n ) n≥1 ; we can then transfer these convergences to L n , writing again, for any t > 0, P(1 + L n ≤ tb n ) = P(τ tbn > n). Recall that we focus on the nullrecurrent case, so δ ∈ [-1, 1], κ ∈ [0, 1]. Let (a n ) n≥1 be a sequence such that P(τ 1 ≥ a n ) ∼ n -1 as n → ∞, then:

(i) Case δ = 1, κ = 1. Then a -1 n (τ nnµ(a n )) converges in distribution to a 1 stable law, see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]IX.8] (see Eq. (8.15) for the centering): in particular, τn nµ(an) converges in probability to 1 (recall that µ(•) is defined in (3.5)).

Setting b n such that b n µ(a bn ) ∼ n, then we have that τ tbn n converges to t in probability, so b -1 n L n converges in probability to 1. Then, we simply have to notice that if b n is given by the above relation then we have µ(a bn ) ∼ µ(n), see e.g. [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF]Lem. 4.3], so b n ∼ n/µ(n) as defined in (3.5).

(ii) Case δ ∈ (-1, 1), κ ∈ (0, 1). Setting a n = Γ(1κ) 1/κ a n we have that 1 an τ n converges in distribution to a one-sided κ-stable random variable Z with Laplace transform e -t κ , see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]XIII.6].

Let us set b n such that a bn ∼ n; using that a n is regularly varying with exponent 1/κ, we get that a tbn ∼ t 1/κ a bn ∼ t 1/κ n. We therefore get that lim n→∞ P(τ tbn > n) = P(Z > t -1/κ ), showing that b -1 n (1 + L n ) converges in distribution to Z κ . Now, recalling the definition of a n and of a n , we get that P(τ 1 ≥ n) ∼ P(τ 1 ≥ a bn ) ∼ Γ(1κ) -1 P(τ 1 ≥ a bn ) ∼ Γ(1κ) -1 b -1 n , which, combined with (5.1), gives the definition of b n in (3.5).

(iii) Case δ = -1, κ = 0. Then n K 0 ν(τn) converges in distribution to an Exp(1) variable, see [START_REF] Darling | The influence of the maximum term in the addition of independent random variables[END_REF]Thm. 8].

Let b n = 1 K 0 ν(n) . The slowly varying function ν is asymtpotically equivalent to a strictly increasing and continuous function, so assume that ν is strictly increasing and continuous. Then, for any t > 0,

P(1 + L n ≤ tb n ) = P(τ tbn > n) = P tb n K 0 ν(τ tbn ) < tb n K 0 ν(n) ---→ n→∞ 1 -e -t ,
which proves the last claim.

It remains to show the uniform bound on the tail of L n . Here again, we simply need a uniform bound on the tail of τ n , of the type P(τ tbn > n) ≤ Ch(t) for some "good" function h. These estimates are standard and fall in the "big-jump" phenomenon: in all the cases we consider, there exists a constant C > 0 such that for any t < 1 (the event τ tbn > n is an upper large deviation) we have P(1 + L n ≤ tb n ) = P(τ tbn > n) ≤ C × tb n P(τ 1 > n) .

(5.2)

If δ = 1, this is given by [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]Thm. 1.2]; note also that in this case lim n→∞ b n P(τ 1 > n) = 0, thanks to [7, Prop. 1.5.9.a.]. If δ ∈ (-1, 1), this is contained in [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]Thm. 1.1]; note that by definition of b n we have that b n P(τ 1 > n) remains bounded in this case. In the case δ = -1, this is due to [START_REF] Nagaev | On sums of independent random variables without power moments[END_REF] (see also [START_REF] Alexander | Local limit theorems and renewal theory with no moments[END_REF]); and b n P(τ 1 > n) also remains bounded in this case, by definition of b n . In all cases, bounding the probability by 1 if t ≥ 1, we get that P(1 + L n ≤ tb n ) ≤ C min(t, 1) uniformly in n.

1 =

 1 , Thm. 1]:Theorem B. If the (ξ i ) i≥1 are i.i.d., then 1 -E s T + exp -∞ n=1 s nn P(S n > 0) for any s ∈ [0, 1).

m

  , with by convention g(m) = 0 if m < 0. Then we have for all n ≥ 1

1 and

 1 Lemma 3.2 do not depend on the (anti-symmetric) function f in the definition (3.1) of ζ n . In particular, the bounds (3.2) obtained in the case of the simple random walk (and in Proposition 3.5 below for Bessel-like random walks) are valid for any anti-symmetric function function f .

  where the (ε i ) 1≤i≤n are i.i.d. signs (i.e. such that P(ε i

  exp uτ 1 1 {τ 1 ≤δ(t)n} tn , with u = 1 δ(t)n . By Markov's inequality, the first term is bounded by tδ(t) -1 E[τ 1 ]. For the second term, we use that e x ≤ 1 + (e -1)x ≤ 1 + 2x for all x ∈ [0, 1]: hence, for u = 1 δ(t)n , we have e -un E exp uτ 1 1 {τ 1 ≤δ(t)n} ≤ e -un 1 + 2uE[τ 1 ] tn ≤ e -1 2 un , where we have also used that (1 + 2uE[τ 1 ]) tn ≤ e

	P	1 2 un since 2tE[τ 1 ] ≤ 1 2 . We therefore end up with

For non-asymptotic bounds, one could also use the following: for m ≥ 1, e -1 6m ≤ √ πmg(m) ≤ e -1

24m .

Acknowledgements. We are grateful for the numerous discussions on this subject with our colleagues from LPSM; we would like to thank in particular Thomas Duquesne, Nicolas Fournier and Camille Tardif.


Mohamed Abdi
email: abdi.mohamed1@live.fr

Meryem Ould-Rouiss

Lalia Abir Bouhenni

Elhouda Beladjine

Abdelkarim Bekhtaoui

Effect of Reynolds Number on Turbulence Characteristics of Turbulent Ostwald de Waele Fluids

Keywords: Large eddy simulation (LES), Shear-thinning, Shear-thickening, Turbulence

The current work aimed to understand how the Reynolds number affected the characteristics of turbulent power-law fluids. A fully developed large-eddy simulation (LES) of turbulent pseudoplastic and dilatant fluids over a straight pipe was performed numerically at three Reynolds numbers (4000, 8000, and 12000) and flow behaviour indexes of (0.75, 1 and 1.4). The key findings indicated that when the Reynolds number rose, the creation of axial fluctuations close to the wall area was amplified significantly. This led to a notably improved kinetic energy of turbulent fluctuations across the radial direction. The mechanism of the velocity variations from the near-wall area towards the core region was enhanced with rising Reynolds numbers. This trend was more evident as the flow behaviour index dropped.

INTRODUCTION

Non-Newtonian flow plays a significant role in the mechanical and engineering sectors, particularly in processing food, pharmaceuticals, polymers, cement, petroleum, and other materials. In the last several decades, research on the power-law fluids in pipe designs has generated a significant quantity of literature, most of which has focused on the turbulent flow of non-Newtonian fluids [START_REF] Metzner | Flow of non-Newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF][START_REF] Metzner | Non-Newtonian Fluid Flow. Relationships between Recent Pressure-Drop Correlations[END_REF][START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF][START_REF] Vidyanidhi | Non-Newtonian Flow in a Rotating Straight Pipe[END_REF][START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF][START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF][START_REF] Malin | the Turbulent Flow of Bingham Plastic Fluids in Smooth Circular Tubes[END_REF][START_REF] Ohta | DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids[END_REF][START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF][START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF]. For the first time, Dodge and Metzner [START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF], in their article, reported a theoretical analysis turbulent flow of non-Newtonian fluids through smooth pipes. With four concentrations of a polymer shear-thinning solution (sodium carboxymethyl cellulose) and water, Pinho et al. [START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF] assessed the mean axial velocity and the three normal stresses in fully developed pipe flow across a range of Reynolds numbers from 240 to 111,000. Direct numerical simulation (DNS) was carried out by Gavrilova and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF] [START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF] at two generalised Reynolds values, 10000 and 20000, across power-law index ranges of 0.4 to 1. Recently, Singh et al. [START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF] used (DNS) at a friction Reynolds number Reτ=323 to examine the impact of the flow index parameter of power-law fluids in turbulent pipe flow.

In a more recent study, Abdi et al. [START_REF] Abdi | Numerical simulation of turbulent forced convection of a power law fluid flow in an axially rotating pipe[END_REF] used LES and an extended Smagorinsky model to perform fully developed turbulent forced convection of thermally independent pseudoplastic fluid via an axially heated rotating conduit. With a rotational speed ranging from 0 to 3, the simulation is working fluid Reynolds and Prandtl values were set to 4000 and 1, respectively. The temperature along the pipe wall radius is shown to fall as the rotation rate rises pipe wall rotates dramatically. This is due to a centrifugal force that substantially causes the mean axial velocity profile to rise when the apparent fluid viscosity in the pipe core region lowers.

In the current study, with a typical dynamic model and the LES technique, a numerical analysis of the fully developed turbulent flow of power-law fluids through a pipe was carried out for three simulations of Reynolds values (Res=4000, 8000, and 12000). The flow behaviour indices used in this research primarily examined pseudoplastic and dilatant fluids were 0.75 (shear-thinning), 1, and 1.4, respectively (shearthickening). The investigation aimed to demonstrate how the Reynolds number affected the turbulence characteristics through analysis and discussion of the key turbulence statistics related to these problems.

GOVERNING EQUATIONS AND NUMERICAL PROCEDURE

Governing Equations

The present study was devoted to a fully developed turbulent flow of pseudoplastic (n=0.75), dilatant (n=1.4) and Newtonian (n=1) fluids through an axially stationary pipe with a computational domain of 20R (Fig. 1) by employing LES with a standard dynamic model. Three simulation Reynolds numbers (Res) were considered (4000, 8000 and 12000), which were based on the centerline axial velocity of the analytical fully developed laminar profile (UCL), which were defined respectively as Res=ρUCL 2-n R n /K and UCL=(3n+1).Ub/(n+1). The filtered equations can be expressed as follows (Equations 1 and 2): The constitutive equation for the power-law model is given by:

0 i i u x  =  (1) 1 1 Re j i j j ij n i i j s i i j i u u u u u dP t x x x x x x   -         + = - + + +              (2)
n K  = (3)
The variables K and n are the fluid consistency index and the flow behaviour index, respectively. The shear-thinning behaviour occurs for 1 n  , shear-thickening for n>1, and n=1 the fluid shows Newtonian behaviour.

Numerical Procedure

The present investigation was performed with a grid resolution of 65 3 grid points in axial, radial and circumferential directions. The governing equations were discretised on a staggered mesh in cylindrical coordinates with a computational length in the axial direction 20R. The numerical integration was performed using a finite difference scheme, second-order accurate in space and time. The time advancement employed a fractional-step method. A third-order Runge-Kutta explicit scheme and a Crank-Nicholson implicit scheme were utilised to evaluate convective and diffusive terms. The above mathematical model was implemented in a finite difference laboratory code. In the non-Newtonian Smagorinsky model, the subgrid stress tensor ij  is linked to the strain rate tensor by 2

ij t ij S  =-
. The turbulent viscosity is computed by ( )

2 t s s n ij C f f S  = ,
where  is the computational filter, Cs the model constant, fs the van Driest wall damping function, and fn is the correction function for the change in viscosity.

RESULTS AND DISCUSSION

The aim was to discern or distinguish the Reynolds number influence on the power-law fluids turbulence characteristics in turbulent pipe flow. The study was focused on the pseudoplastic (shear-thinning n=0.75), dilatant (shear-thickening n=1.4) and Newtonian (n=1) fluids at low (Res=4000), moderate (Res=8000) and high (Res=12000) simulation Reynolds numbers. To ascertain the accuracy and reliability of the results, the turbulence intensities of the axial, radial and tangential velocity fluctuations of a Newtonian fluid compared reasonably with that of Redjem et al. [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] at the Reynolds number of 5500. It was apparent that no significant difference was observed between the RMS profiles of the axial component, where the turbulence intensities of the axial velocity collapsed with that of Redjem et al. [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] along the radial coordinate. The turbulence intensities of the radial and tangential velocity fluctuations were slightly overestimated compared to those of Redjem et al. [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF]. The small discrepancy may have been due to the difference in the Reynolds number value and the numerical solution procedure.

Turbulence Intensities of velocity fluctuations

The root mean square (RMS) distributions of the axial, radial and tangential velocity fluctuations along the pipe radius versus the distance from the wall in wall units Y + scaled by the friction velocity are presented respectively in Fig. 2, Fig. 2 and Fig. 5. As appears in Fig. 2, the turbulence intensities of the axial velocity fluctuations were almost identical over the near-wall region (Y + <3) for all studied cases, which was due to the absence of turbulent shear stress where the molecular shear stress force was dominant in this flow region. Beyond approximately (Y + =5), the profiles increased and deviated from each other gradually the further away from the wall with the wall distance Y + . The axial turbulence intensities began to enhance significantly beyond (Y + =5), which indicated that the axial velocity fluctuations were generated in the near-wall region. There was a clear trend of increasing axial turbulence intensities along the radial out of the viscous sublayer (0≤Y + ≤5). This meant the axial fluctuations propagated from the wall towards the remaining flow regions. Beyond approximately (Y + =20), the axial turbulence intensities began to drop progressively towards the core region. These profiles fell to lower values in the core region for all cases. In other words, the axial velocity fluctuations gradually vanished with the logarithmic region wall distance (30≤ Y + ≤200).

It can be seen from Fig. 2 that the axial turbulence intensities profiles of the Reynolds number of 12000 were considerably larger than those of 8000 and 4000 along the radial direction for all flow behaviour indices, especially at the peak value location. The increased Reynolds number also led to shifting the peak location from the wall towards the core region. It can be said that the increased Reynolds number resulted in a pronounced enhancement in the generation of axial fluctuations near the wall region, which also led to ameliorating the transport of these fluctuations from the viscous sublayer towards the core region along the radial direction. It was worth noting that this trend was more pronounced as the flow behaviour index (n) decreased. On the other hand, the RMS profiles of the shear-thinning fluids lay above those of the shearthickening and Newtonian fluids along the pipe radius. This meant that the decreased flow behaviour index (n) led to improving the generation and the propagation of the axial fluctuations along the radial direction, as pointed out by Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF], where this trend was more pronounced as the Reynolds number increased. As appears in Fig. 3, the turbulence intensities of the radial velocity fluctuations were nearly linear and equal to zero value along the near-wall region (Y + <3) for all cases. It can be said that the RMS of the radial component was nearly independent of the Reynolds number and the flow behaviour index in this region. At a large wall distance, the effects of the Reynolds number and flow behaviour index became more significant with the wall distance. The radial turbulence intensities began to increase gradually far away from the wall with wall distance Y + . Moreover, these profiles began also to deviate from each other out of the viscous sublayer (0≤Y + ≤5), where this deviation became more distinct with increasing Reynolds number. Out of the buffer region (5≤ Y + ≤30), the radial turbulence intensities profiles began to decrease noticeably after reaching their peak values in the logarithmic region (30≤ Y + ≤200) for the three Reynolds numbers.

As shown in Fig. 3, the profiles of the 12000 lay slightly below those of the 8000 and 4000 along the pipe radius. These profiles of 12000 appeared to shift away from the wall towards the core region as compared with those of 8000 and 4000 for all flow behaviour indices. It was evident that the increased Reynolds number led to attenuating or lessening the radial turbulence intensities slightly and also led to improving the propagation of these fluctuations from the wall to the core region. This trend was more pronounced as the flow behaviour index decreased. On the other hand, the pseudoplastic (n=0.75) profiles lie down those of the dilatant (n=1.4) and Newtonian fluid for all Reynolds numbers along the radial coordinate, where the decreased flow behaviour index results in a marked attenuation in the radial turbulence intensities over the pipe radius, as pointed out by Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. Figure 4 shows the distribution of turbulence intensities of the tangential velocity fluctuations. These profiles had the same trend over the pipe radius, where there was a clear trend of oscillations in this profile along the radial direction for Reynolds numbers of 4000, 8000 and 12000 for the three flow behaviour indices. It can be said that the tangential turbulence intensities were less affected by the Reynolds number, where it was apparent that no significant noteworthy differences were found between the profiles of 1200, 8000 and 4000 along the radial direction for the three flow behaviour indices.

Turbulent Kinetic energy

The kinetic energy of turbulent fluctuations of the pseudoplastic (n=0.75), dilatant (n=1.4) and Newtonian (n=1) fluids along the pipe radius (R) are illustrated in Fig. 5. The turbulent kinetic energy profile was linear and remained constant in the near-wall region (Y + <1) for all Reynolds numbers because of the absence of velocity fluctuations in the near-wall region. Far away from the wall, the turbulent kinetic energy began to enhance gradually with the wall distance Y + . This enhancement or improvement was related to augmentation in the turbulence intensities of the axial, radial and tangential velocity fluctuations in this flow region. The turbulent kinetic energy profiles reached their peak values. Beyond approximately (Y + =20), these profiles dropped rapidly to zero in the core region for all cases. This reduction occurred because of the vanishing of the axial, radial and tangential turbulence intensities in this flow region.

Interestingly, the axial velocity fluctuations were generated in the near-wall region (Fig. 2) and transferred to the radial and tangential components. These fluctuations vanished in the core region (Fig. 3, Fig. 4), as pointed out by Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. As shown in Fig. 5 the turbulent kinetic energy profile of the Reynolds number of 12000 lies above those of 8000 and 4000 along the pipe radius, especially in the buffer region. The increased Reynolds number resulted in an enhancement in the generation of the axial fluctuations. The transfer mechanism of these fluctuations to the radial and tangential components consequently ameliorated the kinetic energy of turbulent fluctuations over the radial direction. It should be noted that this trend was more pronounced as the flow behaviour index increased.

Turbulent Reynolds shear stress

The predicted profiles of Reynolds shear stress were equal to zero along the viscous sublayer (0≤ Y + ≤5). These profiles deviated significantly from each other and exhibited a gradual enhancement with wall distance. Beyond the logarithmic region (30≤ Y + ≤200), these profiles dropped to zero in the core region. It was evident that the increased Reynolds number resulted in a marked enhancement in the Reynolds stress, especially in the logarithmic region, for the three flow behaviour indices.

CONCLUSIONS

The current study focused on the numerical analysis of a fully developed turbulent flow of power-law fluids in a conduit using the LES technique and a conventional dynamic model at three Reynolds number simulations (4000, 8000 and 12000). With flow behaviour indices of 0.75 and 1.4, respectively, this research sought to determine the effect of the Reynolds number on the mean turbulence characteristics of turbulent pseudoplastic and dilatant fluids. The results indicated that radial turbulence intensities were attenuated, and the higher Reynolds number significantly enhanced the axial fluctuations near the wall area. Additionally, the higher Reynolds number facilitated the radial transmission of these variations from the viscous sublayer to the core area. These trends became more evident as the flow behaviour index dropped. An increase in Reynolds number enhanced the kinetic energy of turbulent variations in the radial direction. This tendency became more evident as the flow behaviour index rose. Figures list.

Figure 1 :

 1 Figure 1: computational domain

Figure 2 :

 2 Figure 2: Validation of the results.

Figure 3 :

 3 Figure 3: RMS of axial component versus Y + .

Figure 4 :

 4 Figure 4: RMS of the radial component against Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 5 :

 5 Figure 5: RMS of the tangential component versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 6 :

 6 Figure 6: Turbulent kinetic energy against Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 7 :

 7 Figure 7: Turbulent Reynolds stress versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

2

 2 Figures CaptionsFigure 1computational domain Figure 2 Validation of the present results.

Figure 3 RMS

 3 Figure 3 RMS of axial component versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 4 RMS

 4 Figure 4 RMS of the radial component against Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000 Figure 5 RMS of the tangential component versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 6

 6 Figure 6 Turbulent kinetic energy against Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res =8000 and Orange: Res =12000.

Figure 7 Turbulent

 7 Figure 7


Mohamed Abdi
email: abdi.mohamed1@live.fr

Meryem OULD ROUISS b , Manel Ait Yahia

Amina Ould

Large Eddy Simulation Investigation of Reynolds Number Effects on Rheological Behavior of Ostwald de Waele Fluids

Keywords: Large eddy simulation (LES), Shear-thinning, Shear-thickening, Fully developed, Turbulent flow

A fully developed turbulent pipe flow of non-Newtonian fluids was investigated numerically using the large eddy simulation (LES) method. The rheological and hydrodynamic behaviour of the power-law fluids was a primary focus for flow behaviour indexes of 0.75, 1, and 1.4 and at simulation Reynolds numbers of 4000, 8000, and 12000. The calculations utilised a finite difference method with a numerical resolution of 65 3 grid points in the r, θ and z dimensions, respectively. They were second-order accurate in space and time. According to the results, the shear rate profiles throughout the pipe radius for the pseudoplastic and dilatant fluids noticeably decreased and improved with a higher flow index. Additionally, the elevated flow index caused a notable rise in the fluid viscosity close to the wall, which raised the friction factor.

INTRODUCTION

The industry has become very interested in the flow of non-Newtonian fluids since these fluids are used in many technical applications, including the production of cement, food, paper, and paint goods, as well as in the petroleum, pharmaceutical, and polymer processing sectors. A literature review shows that the turbulent flow of non-Newtonian fluids has received significant attention in recent years. Numerous studies have been conducted on the flow of non-Newtonian fluids in axially stationary pipes, using theoretical, experimental, and numerical methods to describe the rheological behaviour and to evaluate the effects of the rheological properties of this type of fluid [START_REF] Metzner | Flow of non-Newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF][START_REF] Metzner | Non-Newtonian Fluid Flow. Relationships between Recent Pressure-Drop Correlations[END_REF][START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF][START_REF] Vidyanidhi | Non-Newtonian Flow in a Rotating Straight Pipe[END_REF][START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF][START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF][START_REF] Malin | the Turbulent Flow of Bingham Plastic Fluids in Smooth Circular Tubes[END_REF][START_REF] Malin | Turbulent pipe flow of Herschel-Bulkley fluids[END_REF][START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF][START_REF] Ohta | DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids[END_REF][START_REF] Rudman | Direct numerical simulation of turbulent non-Newtonian using a spectral element method[END_REF][START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF][START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF][START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF]. Pinho et al. [START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF] used water and four concentrations of a polymer shear-thinning solution (sodium carboxymethyl cellulose) to assess the mean axial velocity and the three normal stresses in fully developed pipe flow in the range of Reynolds numbers from 240 to 111,000. Malin [START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF] employed a modified version of the Lam-Bremhorst K-ɛ model in a turbulent flow to simulate numerically fully developed laminar and turbulent flow of power-law fluids at different Reynolds numbers for various power-law indices. Using the spectral element-Fourier approach, Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF] performed a direct numerical simulation (DNS) of shear-thinning fluids at flow behaviour indices of (0.5, 0.69, and 0.75) More recently, Abdi et al. [START_REF] Abdi | Numerical simulation of turbulent forced convection of a power law fluid flow in an axially rotating pipe[END_REF] used LES and an extended Smagorinsky model to perform fully developed turbulent forced convection of thermally independent pseudoplastic fluid via an axially heated rotating conduit. The simulation Reynolds and Prandtl values for the working fluid were set to 4000 and 1, respectively, with a rotation rate ranging from 0 to 3. The temperature along the pipe wall radius is shown to fall as the rotation rate rises pipe wall rotates dramatically due to a centrifugal force that substantially causes the mean axial velocity profile to rise when the apparent fluid viscosity in the pipe core region lowers. The higher-order statistics show that the rotation rate effect on the skewness and flatness coefficients is primarily limited to the region close to the pipe wall. Further away from the pipe wall, these profiles appear to be almost completely independent of the centrifugal force caused by the rotating pipe wall.

Using a large eddy simulation (LES) and a conventional dynamic model, the current work was focused on a fully developed turbulent flow through an isothermal axially stationary pipe of the power-law fluids. This research examined dilatant (shear-thickening) and pseudoplastic (shear-thinning) fluids spanning the range of 0.75 to 1.4 for different Reynolds values (4000, 8000 and 12000). By examining both rheological properties and mean flow quantities, such as the apparent fluid viscosity, the shear stress, the shear rate, the mean axial velocity, and the friction factor, this study investigated the impact of the Reynolds number on the rheological behaviour numerically and on the flow patterns of this type of non-Newtonian fluid.

GOVERNING EQUATIONS AND NUMERICAL PROCEDURE

Governing Equations

The present study was devoted to a fully developed turbulent flow of pseudoplastic (n=0.75), dilatant (n=1.4) and Newtonian (n=1) fluids through an axially stationary pipe with a computation domain of 20R (Fig. 1), by employing large eddy simulation (LES) with a standard dynamic model. Three simulation Reynolds numbers (Res) were considered in the present study (4000, 8000 and 12000), which was based on the centerline axial velocity of the analytical fully developed laminar profile (UCL), which are defined respectively as Res=ρUCL 2-n R n /K and UCL=(3n+1).Ub/(n+1). Ub is the mean velocity, K is the fluid consistency index, and ReMR is the Reynolds number of Metzner and Reed and is defined as

( ) 2 Re 8 6 2 n nn MR b D U K n  - =+
.

The filtered equations can be expressed as follows (Equations 1 and 2):

0 i i u x  =  (1) 1 1 Re j i j j ij n i i j s i i j i u u u u u dP t x x x x x x   -         + = - + + +              (2)

Numerical Procedure

The governing equations were discretised on a staggered mesh in cylindrical coordinates with a computational length in the axial direction 20R. The numerical integration was performed using a finite difference scheme, second-order accurate in space and time. The time advancement employed a fractionalstep method. A third-order Runge-Kutta explicit scheme and a Crank-Nicholson implicit scheme were used to evaluate convective and diffusive terms. The above mathematical model was implemented in a finite difference laboratory code. The grid 65 3 grid points in axial, radial and circumferential directions, respectively, were found to provide an accurate prediction of the turbulence statistics, in agreement with the available data of the literature, and give a good compromise between the required CPU-time and accuracy.

RESULTS AND DISCUSSION

The rheological properties and mean flow quantities were examined, including the shear rate, apparent viscosity, and mean axial velocity. The friction factor was to explore the influence of the Reynolds number on the rheological and flow patterns of shear-thinning (n<1) and shear thickening (n>1) fluids. The study used LES with a standard dynamic model at three flow behaviour indices (0.75, 1 and 1.4) and three simulation Reynolds numbers (4000, 8000 and 12000).

Figure 2 compares the predicted profile of the turbulent axial velocity of a shear-thinning fluid with a flow index (n) of 0.75 at Reynolds Metzner and Reed number of 4255 with that of DNS data performed by Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF] at Reynolds Metzner and Reed number of 3935. No significant differences were observed between them, where the predicted profile was in good agreement with that of Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF] over the entire flow region.

Mean Normalized Shear Rate and Viscosity

The shear rate and apparent viscosity distributions of the shear-thinning and shear-thickening fluids along the pipe radius are depicted in Fig. 3 and Fig. 4, respectively, against the distance from the wall in wall units Y + . The flow behaviour indexes were set at 0.75, 1 and 1.4 at a simulation Reynolds number of 4000, 8000 and 12000. As shown in Fig. 3 in the viscous sublayer (0≤Y + ≤5), the shear rate profiles were nearly linear, they remained constant along the near-wall region for the three Reynolds numbers. Beyond (Y + =5), the shear rate profiles began to decrease gradually with distance from the wall (Y + ) towards the core region, where this reduction was sharp in the buffer region (5≤Y + ≤30) for all cases.

It was evident that the shear rate profile of the pseudoplastic (n=0.75) lay above those of the Newtonian and dilatant (n=1.4) fluids for all Reynolds numbers, where the decreased flow behaviour index resulted in an enhancement in the shear rate along the radial coordinate. On the other hand, the Reynolds number largely affected the shear rate distributions of the power-law fluids along the pipe radius, especially in the near-wall region, where the shear rate profiles of the Reynolds number of 12000 lie above those of 8000 and 4000. With increasing Reynolds number, the shear rate enhanced significantly along the pipe radius, especially in the viscous sublayer. It was worth noting that this trend was more pronounced as the flow behaviour index decreased. The shear rate of the shear-thinning (n=0.75) fluid was more affected by the Reynolds number than those of the Newtonian and shear-thickening (n=1.4) fluids. As seen in Fig. 4, the apparent viscosity of the shear-thinning (n=0.75) and shear-thickening (n=1.4) fluids were identical and equal to the apparent viscosity at the wall (ηw) in the near-wall region for all Reynolds numbers, where no significant noteworthy differences are observed between the three Reynolds numbers. It can be said that the apparent viscosity of the power-law fluids seemed independent of the flow behaviour index and the Reynolds number along the viscous sublayer. Beyond the buffer region (Y + >5), the apparent viscosity profiles began to deviate from each other significantly further away from the wall, where there was a clear trend of gradually increasing and decreasing in apparent viscosity of the pseudoplastic (n=0.75) and dilatant (n=1.4), respectively with the wall distance (Y + ) towards the core region. The decreased flow behaviour index (n) resulted in a considerable enhancement in the apparent viscosity far away from the wall.

It was apparent in Fig. 4 that the viscosity profiles of the Reynolds numbers 4000, 8000 and 12000 collapsed with each other over the entire flow region, where there was no effect of the Reynolds number on the apparent viscosity distribution of the three flow behaviour indices observed. It can be said that the apparent viscosity of the power-law fluids was nearly independent of the Reynolds number along the radial direction. Figure 5 illustrates the apparent viscosity profile scaled by the viscosity at the wall (ηw) of the shearthinning and shear-thickening fluids against the shear rate scaled by the shear rate at the wall ( w  ). As seen in Fig. 5 the apparent viscosity of dilatant (n=1.4) increased progressively with the shear rate for all Reynolds numbers, as the Reynolds number decreased as the apparent viscosity enhanced rapidly. On the other hand, the apparent viscosity of the pseudoplastic (n=0.75) fluid was inversely proportional to the shear rate: the shear rate increased as the apparent viscosity decreased considerably for the three Reynolds numbers. It should be noted that the apparent viscosity of the shear-thinning fluid attenuated or diminished rapidly as the Reynolds number decreased.

Turbulent axial Velocity Profile

The turbulent axial velocity profile of the pseudoplastic (n=0.75), dilatant (n=1.4) and Newtonian fluids along the pipe radius scaled by the friction velocity against the distance from the wall in wall units Y + is presented in Fig. 6, at simulation Reynolds numbers of 4000, 8000 and 12000. The dash lines represented the universal velocity distributions in the viscous sublayer (0≤Y + ≤5) and in the logarithmic layer (30≤Y + ≤200). Results showed that in the viscous sublayer, the predicted turbulent axial velocity profile of the three flow behaviour indices (n) was almost identical and agreed with the universal linear law U + =Y + up to approximately (Y + =5) for all Reynolds numbers. It can be said that the turbulent velocity was almost independent of the flow behaviour index and the Reynolds number in the near-wall region. Beyond the buffer region (Y + >5), the turbulent axial velocity profiles began to deviate significantly from each other with distance from the wall (Y + ), where this deviation became more distinct the further away from the near-wall region. In the logarithmic layer (30≤Y + ≤200), the profiles of the shear-thinning (n=0.75) fluids lay above those of Newtonian and shear-thickening (n=1.4) fluids for all Reynolds numbers, where this trend was more pronounced as the Reynolds number increased.

It can be said that the turbulent axial velocity was affected largely by the flow index and the Reynolds number out of the buffer region (Y + >30), where the decreased flow behaviour index led to an enhancement of the apparent viscosity, consequently in the turbulent axial velocity in the logarithmic region for the three Reynolds numbers. Moreover, as the Reynolds number increased, the turbulent velocity enhanced considerably in the logarithmic region, where this improvement was due to the increase in the inertia force. Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res=8000 and Orange: Res=12000.

Friction factor

The friction factor of the pseudoplastic (n=0.75), dilatant (n=1.4) and Newtonian fluids at simulation Reynolds numbers of 4000, 8000 and 12000 are shown in Table 1. The friction factor enhanced gradually with the increase in the flow behaviour index. There was a clear trend of a decrease in the friction factor with an increasing Reynolds number. It was worth noting that the increased Reynolds number induced a marked reduction in the friction factor for the three flow behaviour indices, where this was ascribed to the attenuation in the viscous force near the wall region.

CONCLUSIONS

The numerical analysis of a fully developed turbulent flow of pseudoplastic and dilatant fluids in an axial pipe using LES with a conventional dynamic model was presented. This was carried out for three distinct flow behaviour indices (n=0.75, 1 and 1.4) at three simulated Reynolds number levels (Res=4000, 8000, and 12000). This research examined how the Reynolds number affected hydrodynamic and fluid rheological characteristics to characterise this fluid's rheological behaviour and flow patterns in such a situation. The results showed that the Reynolds number rose, and the shear rate of the shear-thinning and thickening fluids was mainly influenced by the Reynolds number over the whole flow zone, which led to an improvement in the shear rate profile along the radial coordinate. When Re had no apparent influence on the viscosity of the three flow behaviour indices, it was concluded that the apparent viscosity of the power-law fluids was almost independent of Re. The turbulent axial velocity in the logarithmic region for the three Reynolds numbers increased in apparent viscosity due to the lowered flow behaviour index. This tendency became more obvious as the Reynolds number rose. Lastly, an increasing Reynolds number significantly attenuated the friction factor for the three flow behaviour indices.

Figure 1 :

 1 Figure 1: computational domain

Figure 2 :

 2 Figure 2: Validation of the results.

Figure 3 :

 3 Figure 3: Shear rate profile versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res=8000 and Orange: Res=12000

Figure 4 :

 4 Figure 4: Apparent viscosity profile versus Y + . Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res=8000 and Orange: Res=12000.

Figure 5 :

 5 Figure 5: Apparent viscosity against the shear rate. Open symbols: n=0.75; Solid line: n=1 and Closed symbols: n=1.4 Red: Res=4000; Blue: Res=8000 and Orange: Res=12000.

Figure 6 :

 6 Figure 6: Axial velocity profile versus Y + .

Table 1 :

 1 Friction factor

	Res		4000			8000			12000
	n	0.75	1	1.4	0.75	1	1.4	0.75	1	1.4
	f.E-1 0.0845 0.107 0.125 0.065 0.0837 0.0977 0.0539 0.0716 0.0833

Red: Res=4000; Blue: Res=8000 and Orange: Res=12000.

Figures list.

Tables Captions

Table 1 Friction factor

Tables list.