Hans Henrik Rugh

Damien Thomine

RATIO ERGODIC THEOREMS: FROM HOPF TO BIRKHOFF AND KINGMAN

Hopf's ratio ergodic theorem has an inherent symmetry which we exploit to provide a simplification of standard proofs of Hopf's and Birkhoff's ergodic theorems. We also present a ratio ergodic theorem for conservative transformations on a σ-finite measure space, generalizing Kingman's ergodic theorem for subadditive sequences and generalizing previous results by Akcoglu and Sucheston.

Introduction and statement of results

Birkhoff's pointwise ergodic theorem [START_REF] Birkhoff | A simple proof of the ratio ergodic theorem[END_REF] is a key tool in ergodic theory. It admits many notable generalizations, including Hopf's ratio ergodic theorem [START_REF] Hopf | Ergodentheorie[END_REF], Kingman's subadditive ergodic theorem [START_REF] Kingman | The ergodic theory of subadditive stochastic processes[END_REF], and more recently Karlsson-Ledrappier and Gouëzel-Karlsson theorems on cocycles of isometries [START_REF] Karlsson | On laws of large numbers for random walks[END_REF][START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF]. Since the work of Kamae [START_REF] Kamae | A simple proof of the ergodic theorem using nonstandard analysis[END_REF] and Katznelson and Weiss [START_REF] Katznelson | A simple proof of some ergodic theorems[END_REF], there exist very short and easy proofs of Birkhoff's ergodic theorem. These proofs have also been adapted to Hopf's and Kingman's ergodic theorems ( [START_REF] Kamae | A simple proof of the ratio ergodic theorem[END_REF] and [START_REF] Katznelson | A simple proof of some ergodic theorems[END_REF][START_REF] Steele | Kingman's subadditive ergodic theorem[END_REF], respectively).

In this article, we provide a proof of Hopf's and Kingman's ergodic theorem, in the context of conservative transformations preserving a σ-finite measure. We follow the argument of Katznelson and Weiss [START_REF] Katznelson | A simple proof of some ergodic theorems[END_REF] but add a noticeable twist: the statement of the ratio ergodic theorem has a natural symmetry which is not present in Birkhoff's ergodic theorem, a symmetry which can be leveraged to simplify proofs. This makes, in our opinion, Hopf's theorem more fundamental, with Birkhoff's theorem now appearing as a corollary (the inverse point of view is given in e.g. [START_REF] Zweimüller | Hopf 's ratio ergodic theorem by inducing[END_REF], where Hopf's theorem is deduced from Birkhoff's by inducing).

As for the Kingman ratio ergodic theorem on a σ-finite measure space, a similar result was obtained by Akcoglu and Suchestom [START_REF] Akcoglu | A ratio ergodic theorem for superadditive processes[END_REF] under an additional integrability assumption. Our result does not make this assumption and the proof is significantly simpler (in our opinion). As the reader may note there are no significant complications coming from working with σ-finite measures, but some parts may be simplified quite a lot if one assumes ergodicity. Definition 1.1. Consider a σ-finite measure space (X, B, µ) and a measure preserving transformation T : X → X. The transformation T is said to be conservative if:

∀A ∈ B : µ(A) > 0 ⇒ ∃n ≥ 1 : µ(A ∩ T -1 A) > 0. (1) A subset A ⊂ X is T -invariant if T -1 A = A, and a function f : X → R is T - invariant if f • T = f . The transformation T is ergodic (for µ) if for any measurable T -invariant subset A, either µ(A) = 0 or µ(X \ A) = 0.
Unless stated otherwise, we make throughout the standard assumption that (X, B, µ, T ) is a conservative measure-preserving transformation on a σ-finite measure space. Most of the time, ergodicity shall not be assumed.

Given any f : X → R, we write S n f := n-1 k=0 f • T k for the Birkhoff sums. Our first goal is to give a proof of the following well-known: Theorem 1.2 (Hopf's ratio ergodic theorem). Let (X, B, µ, T ) be a conservative, measure preserving transformation on a σ-finite measure space. Let f , g ∈ L 1 (X, µ) with g > 0 almost everywhere. Then the following limit exists µ-almost everywhere:

h := lim n→∞ S n f S n g . ( 2
)
The function h is finite µ-almost everywhere, T -invariant, and for any T -invariant subset A ∈ B:

A f dµ = A hg dµ. (3)
Note that, when f > 0 as well, there is a natural symmetry between f and g, which we will exploit in our proof. This symmetry is lost in Birkhoff's version, where g ≡ 1.

We will proceed to prove a ratio version of Kingman's theorem for subadditive sequences. Recall that a sequence (a n ) n≥1 of measurable functions is said to be subadditive (with respect to T ) if for n, m ≥ 1, we have µ-almost everywhere:

a n+m ≤ a n + a m • T n . ( 4
)
Theorem 1.3 (A Kingman ratio ergodic theorem). Let (X, B, µ, T ) be a conservative, measure preserving transformation on a σ-finite measure space. Let (a n ) n≥1 be a sub-additive measurable sequence of functions with values in [-∞, +∞], and with (a 1 ) + ∈ L 1 (X, µ). Let g ∈ L 1 (X, µ) with g > 0 almost everywhere. Then the following limit exists µ-almost everywhere:

h := lim n→+∞ a n S n g ∈ [-∞, +∞). If A ∈ B is a T -invariant set, then inf n 1 n A a n dµ = lim n 1 n A a n dµ = A hg dµ ∈ [-∞, +∞).
For a similar version of this theorem, but under an additional uniform integrability condition on a n , cf. [START_REF] Akcoglu | A ratio ergodic theorem for superadditive processes[END_REF]. To our knowledge the theorem is new in the stated generality.

The remainder of this article is organized as follows. In Section 2 we prove some classical lemmas about conservative dynamical systems, and the main lemma (Lemma 2.3, which slightly generalizes the main theorem of [START_REF] Katznelson | A simple proof of some ergodic theorems[END_REF]). In Section 3 we prove Hopf's ratio ergodic theorem, and in Section 4 the above Kingman ratio ergodic theorem (whose proof uses Hopf's ergodic theorem).

Main lemmas

First, and so that our proofs will be essentially self-contained, let us state and prove some consequences of conservativity. For details the reader may consult e.g. [1, Chap 1]. Conservativity is an a priori mild recurrence condition which is equivalent to the following seemingly stronger recurrence condition. Let (X, B, µ, T ) be a conservative, measure-preserving dynamical system. Then, given any measurable subset A, almost everywhere on A,

n≥0 1 A • T n = +∞. ( 5
)
In other words, almost every point in A returns infinitely often to A. A consequence is the following:

Lemma 2.1. Let (X, B, µ, T ) be a measure-preserving and conservative transformation. Let f : X → R + be measurable. Then, µ-almost everywhere on {f > 0}:

lim n→+∞ S n f = +∞. (6) Proof. For n ≥ 1, let A n := {f ≥ 1 n }. Let Ω n := A n ∩ m≥0 k≤m T -k
A n be the set of points of A n which return to A n infinitely many times.

Since f is positive and takes value at least 1/n on A n , Equation ( 6) follows for all

x ∈ Ω n . Let Ω ∞ := n≥1 Ω n . Then Equation (6) holds on Ω ∞ . Since (X, B, µ, T ) is assumed to be conservative, µ(Ω n ∆A n ) = 0, so that µ(Ω ∞ ∆ n≥1 A n ) = 0. But, n≥1 A n = {f > 0}, so Ω ∞ has full measure in {f > 0}.
Let (a n ) n≥1 be a super-additive sequence of functions and g ∈ L 1 (A, µ). We define for every x ∈ X the following lower and upper limits:

0 ≤ h(x) = lim inf n→∞ a n (x) S n g(x) ≤ h(x) = lim sup n→∞ a n (x) S n g(x) ≤ +∞. ( 7
)
Both h and h are measurable and, in fact, a.e. T -invariant:

Lemma 2.2. Let (X, B, µ, T ) be a measure-preserving and conservative transformation. Let (a n ) n≥1 be a super-additive sequence of functions, with a 1 ≥ 0 a.e.. Let g ∈ L 1 (A, µ; R * + ). Then h • T = h and h • T = h almost everywhere.

Proof. We prove the result for h; the proof for h is essentially the same. Let (a n ) and g be as in the lemma. Then:

a n+1 S n+1 g ≥ a 1 + a n • T g + (S n g) • T
By Lemma 2.1, lim n→+∞ S n g = +∞ almost everywhere, whence, taking the liminf,

h ≥ h • T.
The function η = h/(1 + h) takes values in [0, 1] and we have

A := {h > h • T } = {η > η • T }. By Lemma 2.1, the map φ := η -η • T verifies lim n→+∞ S n φ = +∞ almost everywhere on A. But as S n φ = η -η • T n+1 ∈ [-1, 1] everywhere, we must have µ(A) = 0.
We can now state and prove our main lemma.

Lemma 2.3. Let (X, B, µ, T ) be a measure-preserving and conservative transformation. Let (a n ) n≥1 be a super-additive sequence of functions, with a 1 ≥ 0 almost everywhere. Let g ∈ L 1 (A, µ; R * + ).

Then h is T -invariant, and for all T

-invariant A ∈ B, lim inf n→+∞ 1 n A a n dµ ≥ A hg dµ. ( 8
)
Proof. By Lemma 2.2, h is T -invariant, but with values a priori in [0, +∞]. For ε > 0, we set

h ε := h 1 + εh ,
with the convention that h ε (x) = 1/ε when h(x) = +∞. Let:

n ε := inf k ≥ 1 : a k ≥ S k (h ε g) .
Then n ε (x) = 1 whenever h(x) = 0. When h(x) > 0, we have h ε (x) < h(x), so n ε (x) is finite by the definition of the limsup. Introduce a time-cutoff L ≥ 1 and denote E = E ε,L := {n ε ≤ L}. We then set:

ϕ(x) = ϕ ε,L (x) := 1 if x / ∈ E n ε (x) if x ∈ E .
One verifies that for every x ∈ X:

a ϕ(x) ≥ S ϕ(x) (gh ε 1 E ).
When x ∈ E this is true by the very definition of n ε (x), while for x / ∈ E it holds because the right hand side vanishes.

Define a sequence of stopping times:

τ 0 (x) = 0, τ k+1 (x) = τ k (x) + ϕ T τ k (x) x , k ≥ 0.
.

Note that 1 ≤ τ k+1 -τ k ≤ L for every k ≥ 0, and for all x ∈ X:

a τ k (x) (x) ≥ k-1 j=0 a ϕ(x) • T τj (x) ≥ k-1 j=0 S ϕ(x) (h ε g1 E ) • T τj (x) = S τ k (x) (gh ε 1 E )(x).
Let N ≥ 1 and x ∈ X. There exists k ≥ 1 such that N < τ k (x) ≤ N + L. Then:

a N +L (x) ≥ a τ k (x) (x) + N +L i=τ k (x)+1 a 1 • T i ≥ S τ k (x) (h ε g1 E )(x) ≥ S N (gh ε 1 E )(x),
which allows us to get rid of the intermediate stopping times. Take now a Tinvariant set A ∈ B and integrate the above inequality over A. By T -invariance:

1 N + L A a N +L dµ ≥ 1 N + L A S N (h ε g1 E ) dµ = N N + L A h ε g1 E dµ.
Letting N → +∞, we conclude that:

lim inf n→+∞ 1 n A a n dµ ≥ A h ε g1 E dµ.
Letting L → +∞ and finally ε → 0, we obtain by monotone convergence:

lim inf n→+∞ 1 n A a n dµ ≥ A hg dµ.

Proof of Hopf's Ratio ergodic theorem

We are now ready to prove to prove Hopf's ergodic theorem. Lemma 2.3 only provides a upper bound on h. In most proofs using these techniques, the lower bound on h follows by repeating the same argument, with a modified stopping time (and some handwaving). Here we notice that the symmetry of Hopf's ratio ergodic theorem provides us with a shortcut:

Proof of Theorem 1.2. Let f , g ∈ L 1 (X, µ; R * + )
. By Lemma 2.3, with a n = S n f , for any T -invariant measurable A:

A f dµ ≥ A hg dµ.
In particular, taking A = X, we see that hg ∈ L 1 (X, µ).

We now use the symmetry, and apply Lemma 2.3 with g and f . Since

lim sup n→+∞ (S n g)/(S n f ) = h -1 ,
we get h -1 f ∈ L 1 (X, µ), and in particular +∞ > h ≥ h > 0 almost everywhere.

We now apply Lemma 2.3 again, with hg and f . Since

lim sup n→+∞ S n (hg) S n f = h h ,
we get that, for any T -invariant measurable A,

A f dµ ≥ A hg dµ ≥ A f h h dµ.
As f > 0 a.e. and the integral is finite we conclude that h = h =: h almost everywhere.

Let us now turn towards the proof of Theorem 1.2 without positivity assumption on f . Since µ is σ-finite, we can write f = f + -f -, with f + and f -in L 1 (X, µ; R * + ). Then, µ-almost everywhere:

lim n→+∞ S n f S n g = lim n→∞ S n f + S n g - S n f - S n g = h + -h -=: h. In addition, A f dµ = A (f + -f -) dµ = A (h + -h -)g dµ = A hg dµ.
Remark 3.1. The proof of Theorem 1.2 itself can be significantly shortened if one assumes that (X, B, µ, T ) is ergodic: since h and h are then constant, applying Lemma 2.3 to the pairs (f, g) and (g, f ) yields directly:

h ≥ X f dµ X g dµ ≥ h.
There is, to our knowledge, not much gain to be had in the proof of Lemma 2.3.

In the ergodic case, the statement of Hopf's ergodic theorem can be simplified.

Corollary 3.2 (Hopf's theorem, ergodic version). Let (X, B, µ, T ) be a measurepreserving, conservative and ergodic transformation. Let f , g ∈ L 1 (A, µ) with X g dµ = 0. Then µ-almost everywhere:

lim n→+∞ S n f S n g = X f dµ X g dµ . (9)
Proof. We decompose as above f = f + -f -, with f ± integrable and positive. By the Theorem 1.2, µ-almost everywhere,

lim n→+∞ S n g S n f ± = k ± .
By ergodicity, the k ± are constant and then non-zero, since X g dµ = k ± X f ± dµ = 0. Thus, almost everywhere:

lim n→∞ S n f S n g = lim n→∞ S n f + S n g - S n f - S n g = 1 k + - 1 k - = X (f + -f -) dµ X g dµ = X f dµ X g dµ .
As a special case, we may also consider when µ is probability measure and g ≡ 1 (thus integrable). T is automatically conservative by Poincaré recurrence theorem. From Theorem 1.2 we deduce: Corollary 3.3 (Birkhoff's Ergodic Theorem). Let (X, B, µ, T ) be a measure preserving transformation on a probability space. Let f ∈ L 1 (X, µ). Then the following limit exists µ-almost everywhere:

f * := lim n→+∞ 1 n S n f. ( 10
)
f * is T -invariant (up to a set of measure 0), and for any T -invariant measurable subset A:

A f dµ = A f * dµ. ( 11
)

Kingman, σ-finite version

We proceed here with a ratio version of Kingman's theorem for non-negative super-additive sequences, from which Theorem 1.3 shall follow easily.

Proposition 4.1. Let (X, B, µ, T ) be a measure-preserving and conservative transformation. Let (a n ) n≥1 be a super-additive sequence of functions, with a 1 ≥ 0 almost everywhere. Let g ∈ L 1 (A, µ; R * + ). Then the following limit exists µ-almost everywhere:

h := lim n→+∞ a n S n g ∈ [0, +∞].
In addition, for any T -invariant measurable set A,

sup n∈N 1 n A a n dµ = lim n→+∞ 1 n A a n dµ = A hg dµ ∈ [0, +∞].
Proof. Let A be any T -invariant measurable set. By Lemma 2.3, we know that:

lim inf n→+∞ 1 n A a n dµ ≥ A hg dµ.
We want to prove the converse inequality (inverting the direction of the inequality, and the lim inf and lim sup). Let K < sup n≥1

1 n A a n dµ. Then we can find k ≥ 1 such that 1 k A a k dµ > K. For M > 0, let f k,M := min{a k , M S k g, M k}/k.
By the monotone convergence theorem, there exists M > 0 such that:

A f k,M dµ > K.
Let n ≥ 2k, and let q, r be such that n = qk + r and 0 ≤ r < k. Then:

a n = k-1 i=0 a n k ≥ k-1 i=0 1 k   a i + q-2 j=0 a k • T i+jk + a n-i-(q-1)k • T i+(q-1)k   ≥ (q-1)k-1 i=0 a k k • T i = S (q-1)k (a k /k) ≥ S n-2k f k,M . Let h k,M := lim inf n→+∞ S n f k,M /S n g. Note that |S n f k,M -S n-2k f k,M | ≤ 2kM , so that h k,M = lim inf n→+∞ S n-2k f k,M /S
A hg dµ = A hg dµ. ( 12
)
All is left is to prove that h = h almost everywhere. For M ≥ 0, take A := {h ≤ M }.

Then hg is integrable on A, and since g > 0 a.e. Equation ( 12) implies h = h almost everywhere on A. Since this is true for all M ≥ 0, we get that h = h almost everywhere on {h < +∞}, and obviously h = h on {h = +∞}.

Let us finish the proof of Theorem 1.3.

Proof of Theorem 1.3. Up to taking the opposite sequences, we work with superadditive sequences. Let (a n ) n≥1 be a super-additive sequence, and g a positive and integrable function. Write a 1 = a + 1 -a - 1 and b n := a n + S n a - 1 . Then (b n ) n≥1 and g satisfy the hypotheses of Proposition 4.1, and so do (S n a - 1 ) n≥1 and g. The (almost everywhere) limits and integrals concerning (S n a - 1 ) n≥1 and g are finite, so we can subtract them from the limits and integrals concerning (b n ) n≥1 and g.

  n g by Lemma 2.1. Hence, h ≥ h k,M . By Hopf's theorem (cf. Theorem 1.2),

	K ≤
	Since this is true for all K < sup n≥1	1 n A a n dµ, we finally get:
	lim sup n→+∞	1 n A	a n dµ ≤ sup n≥1	1 n A	a n dµ ≤	A	hg dµ,
	whence the sequence ( 1 n A a n dµ) n≥1 converges to its supremum, and:

A f k,M dµ = A h k,M g dµ ≤ A hg dµ.


Bernard Picinbono
email: bernard.picinbono@lss.supelec.fr

Singularity in signal theory

Keywords: Statistical signal analysis, Signal and noise modeling, Non-Gaussian signals and noise, Markov processes., Cantor sets, spectral measure

A discrete-time random signal is singular if its values are singular random variables defined by a distribution function continuous but with a derivative equal to zero almost everywhere. Singular random signals can be obtained at the output of some linear filters when the input is a discrete-valued white noise. Sufficient conditions for singularity are established. In particular it is shown that if the poles of the filter are inside a circle called the circle of singularity and if the input is white and discrete-valued the output is singular. Computer experiments using histograms at different scales exhibit the structure of singular signals. The influence of input correlation is also analysed. It is shown that when the input is not white, but has a specific Markovian structure, the output can be singular. This is also verified by computer experiments. Finally, mixtures of singular and discrete-valued random signals are analysed.

I. INTRODUCTION

Singularity is a concept introduced in probability theory and related to properties of distribution functions of random variables. Usually only two kinds of random variables are considered: those that are continuous and having a distribution function (DF) with a derivative (probability density function) and those that are discrete with a DF varying by steps defining possible values and their corresponding probabilities. However there is a third kind of random variables: those that are singular.

A random variable (RV) is said to be singular if its DF is continuous but with a derivative equal to zero almost everywhere. Then such a RV is neither continuous (no probability density function) nor discrete because its DF is continuous and not a stepwise function.

Singular RVs are often considered as mathematical curiosities without interest in signal theory and engineering sciences. They are rarely introduced in standard textbooks [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF][START_REF] Ochi | Applied Probability and Stochastic Processes[END_REF][START_REF] Pfeifer | Probability for Applications[END_REF][START_REF] Helstrom | Probability and Stochastic Processes for Engineers[END_REF][START_REF] Picinbono | Random Signals and Systems[END_REF]) and appear only in mathematically oriented books (Loève 1977, [START_REF] Wong | Stochastic Processes in Engineering Systems[END_REF]. Very simple RVs however can be singular. The best example comes from an old result known for more than sixty years. It says that if w k is a set of independent and identically distributed (IID) RVs taking the values ±1 with the same probability, then the sum of the series ∞ k=0 a k w k is singular as soon as |a| < 1/2. So one of the simplest RV that can be considered is singular. This result is often omitted because its usual proof requires very abstract reasoning. One of the first tasks of this paper is to show its origin and to introduce an elementary proof that can afterwards be used in signal theory. Indeed the series considered above is similar to the output of an exponential discrete-time filter whose input is a discrete valued white noise. This is the case of autoregressive signals widely used in signal processing.

Then the question of knowing whether or not this can be extended to other class of signals and systems appears immediately.

The main result of this paper is that singularity of the output of a linear filter depends on two facts: the discrete character of the input, which is a common situation in communication theory, and some specific properties of the filter such as the location of its poles in a circle called the circle of singularity. In order to provide a better understanding of a problem usually widely ignored, and to visualize how singularity can appear for rather simple signals, number of computers experiments are presented.

In the last part of the paper it is also shown that the assumption of whiteness, fundamental in the proof of the basic result, can be partially deleted and we present some examples of coloured input signals leading also to singular outputs. This is especially the case of some Markov processes.

II. STATEMENT OF THE PROBLEM AND REVIEW OF KNOWN RESULTS

Let X(ω) be a RV defined on some probability space. Let F (x) be its distribution function (DF). In all of what follows we do not make any difference between two RVs distinct but equal with probability 1. This means that X(ω) is entirely defined by its DF F (x). The RV X(ω) is said to be singular if its DF F (x) is continuous but with a derivative equal to zero almost everywhere.

At the beginning let us remind some fundamental results of probability theory used in the discussion that follows and especially the Lebesgue decomposition theorem [START_REF] Lukacs | Characteristic Functions[END_REF]. For this let us recall that a RV is said to be continuous if its DF is continuous and has a derivative which is its probability density function (PDF). On the other hand a RV is said to be discrete if its DF varies only by steps of amplitudes p i at points x i . This means that the RV takes only the values xi with the probabilities p i .

The Lebesgue theorem says that any DF F (x) can be decomposed in an unique way as a sum of three terms, or that

F (x) = a 1 F c (x) + a 2 F d (x) + a 3 F s (x), a i ≥ 0, a 1 + a 2 + a 3 = 1. (1)
In this equation the three functions F i (x) are DFs and F c(x), F d(x), and F s(x) are the continuous, discrete and singular components of F (x) respectively. If a 1 = 1, the RV is continuous and its PDF is

the derivative of F c (x). If a 2 = 1, X is a discrete RV. If, finally, if a 3 = 1, the RV X is singular. If
one of the coefficients a i is equal to 1, the DF is said to be pure. The spectrum S F of a RV is the set of the points of increase of its DF F (x) and the spectral measure (SM) is the Lebesgue measure L(S F ) of this set. It is clear that if the SM is zero the continuous part in (1) is zero, or a 1 = 0. Then the RV can be discrete (a 3 = 0), singular (a 2 = 0), or a mixture of a discrete and a singular parts. Then to show that a RV is singular it suffices to show that its SM is zero and that there is no discrete part in the decomposition of its DF.

The fundamental theorem opening all this discussion is the following. Consider a set of RVs w k independent and identically distributed (IID) with a symmetric Bernoulli distribution. This means that the w k s take only two values with the same probabilities 1/2. When this is not otherwise explicitly indicated the two possible values of each RV w k are ±1. Consider the RV X defined by the series

X = ∞ k=0 a k w k , (2)
which is convergent provided that |a| < 1. Note that the possible values of w k imply that X is symmetric, which means that X and -X have the same DF. Furthermore, for the same reasons, changing a in -a does not change the DF of X. Then we can assume that a ≥ 0. The following results holds:

February 22, 2018 DRAFT

(1) if a < 1/2, X is singular;

(2) if a = 1/2, X is uniformly distributed in [-1/2, +1/2]; For this purpose consider the partial sum X N and the rest R N defined by

(3) if 1/2 < a = 1,
X N = N k=0 a k w k ; R N = ∞ k=N +1 a k w k = a N +1 ∞ k=0 a k w N +k+1 (3)
The RV

X N takes 2 N +1 distinct values v N i with the same probability 1/2 N +1 . The rest R N satisfies |R N | ≤ a N +1 /(1-a), this limit being obtained if all the w N +k+1 are equal to +1 or -1. As a consequence the SM of RN is smaller than a N +1 [2/(1 -a)]. Since there are 2 N +1 distinct values v N i of XN, the SM of X is smaller than (2a) N +1 [2/ 1 -a)]
). As this is valid for all N , this SM is zero whenever a < 1/2.

Note that this property of the SM is due to the value of a and to the fact that the input w k has only two possible values. On the other hand the probabilities of these outcomes do not play any role. We shall see later that this is general. This result means that there is no continuous part in the decomposition of the DF of X, or that a 1 = 0 in (1). Let us see now that there is also no discrete component, or a 2 = 0. Indeed suppose that this is not the case. This would mean that there is a value x 0 such that the probability that X = x 0 is positive. But as x 0 is a value of X, there exists a set of numbers η k taking only the values ±1 such that

x 0 = ∞ k=0 a k η k . Furthermore, as the values v N i are distinct, this set is unique. Since the RVs w k are IID and P[w k = η k ] = 1/2, we deduce P[X = x 0 ] = ∞ k=0 P[w k = η k ] = 0. ( 4
)
As a consequence the RV X is singular, or a 3 = 1. On the contrary to the property of the SM, it is clear that this proof uses the fact that the w k s are independent and that their two values have the same probability 1/2. It is simple to see however that the result remains valid if one of the values has the probability p, except when p = 0 or p = 1, which corresponds to a situation where the input signal w k is no longer random.

DRAFT February 22, 2018

For the discussion that follows it is important to understand that the singularity comes from two completely distinct properties. The first one depends only on a and on the fact that w k takes only two values ±1. But it is insufficient to imply singularity and we shall see later that for some particular probability distributions of the w k the sum X can be discrete or a mixture of discrete and singular parts.

The second properties arises from the whiteness of w k . It is the combination of these two properties which ensures that X is singular.

Let now present an interpretation of X N by its tree of construction presented in figure 1. For each value v N i of X N we can associate the value -v N i . It is obtained simply by changing the signs of the w k s appearing in [START_REF] Garsia | Entropy and singularity of infinite convolutions[END_REF]. This means, as noted above, that the the RV X is symmetric. As a consequence we can consider only positive v N i s For the same reason it is always possible to assume that a > 0. With these assumptions we have v 0 0 = 1. The two positive values of v 1 1 are 1 -a and 1 + a. The construction of the 8 positive values of X 3 appears in the tree of figure 1. Let us now see that the assumption a < 1/2 means that there is no crossing of the branches of the tree. Indeed consider the two branches of the tree starting from a point v N i . There is no crossing between all the branches starting from this point if

v N i -a N +1 + a N +2 /(1 -a) < v N i + a N +1 a N +2 /(1 -a)
. This yields a < 1/2. It is clear that the construction of this tree is similar to the one of Cantor sets. This is why it is sometimes said that the RV X has a Cantor-type distribution [START_REF] Wittke | Infinite series of interference variables with Cantor-type distributions[END_REF]).

Singularity is not limited to random geometric series like (2) but can appear with RVs such as

X = ∞ k=0 h k w k , (5)
where h k > 0 and the w k s have the same properties as in [START_REF] Falconer | Fractal Geometry Mathematical Foundations and Applications[END_REF]. In this case X N and R N of (3) are written simply by replacing a k by k k . The last equality of (3) does not hold.

It is shown without complete proof in p. 66 of [START_REF] Lukacs | Characteristic Functions[END_REF]

) that if ρ = ∞ k=n+1 h k < h n , ∀n, (6)
then X is singular. In the case where k k = a k , this yields a < 1/2. It is possible to construct a tree as in figure 1 with the h k s instead of the a k s. One can then see that the condition (6) implies that there is no crossing between the branches of the tree, which introduces again a Cantor structure [START_REF] Picinbono | Singular random signals[END_REF].

Before leaving this section let us present a short review of some papers from the engineering literature where the problem of singularity is discussed. The treatment of sequences of Bernoulli RVs appear frequently in the context of digital communications. The first discussion concerning consequences of singularity was presented in [START_REF] Hill | Random geometric series and intersymbol interference[END_REF]. The discussion was limited to geometric series like February 22, 2018 DRAFT

(2) and the purpose was to obtain an approximation of the DF for the calculation of the error probability or the performance of communication systems. Extensions of the same problem to Cantor-type distribution was presented in [START_REF] Wittke | Infinite series of interference variables with Cantor-type distributions[END_REF]. In this paper the condition ( 6) is explicitly used and various examples of filters satisfying this condition are introduced. However, as indicated by the authors, the condition h k > 0 used in [START_REF] Hill | Random geometric series and intersymbol interference[END_REF] and introducing the Cantor structure is very restrictive and it is not satisfied by a large class of filters containing for example terms like a k cos(ωk). The principle of calculation of the distribution function and of some expectations is then presented and used for the evaluation of the error probability. A rather more theoretical approach of the same problem is presented in [START_REF] Smith | Error probabilities on fading channels with intersymbol interference and noise[END_REF]). Finally other calculations of expectations of singular RVs are discussed in [START_REF] Campbell | Distributions and expectations of singular random variables[END_REF].

In this paper the singularity is introduced from some properties of the entropy of the RVs by using an approach introduced in ( The first purpose of the present paper is to show that singularity can be introduced from considerations of properties of the poles of a linear filter. Furthermore in all these papers it is assumed that the RVs wk are independent, and this assumption is a corner stone for the introduction of singularity. Then it arises immediately the question of knowing whether or not it can be relaxed without suppressing singularity.

This question is discussed in the second part of the paper.

III. SINGULARITY AND LINEAR FILTERING

The previous discussion has an immediate application in the case of some signals obtained by linear filtering and especially auto-regressive signals. A signal x k is said to be auto-regressive of order 1 [AR(1)] if it is deduced-from a white noise w k by the linear filter defined in the time domain by the first order recursion x k = ax k-1 + w k . It is then defined by the parameter a called the regression coefficient and by the DF of w k . This correspondoing input-output relationship is

x n = ∞ k=0 a k w n-k (7)
It is then a discrete time causal filter with the impulse response a k . If the input signal is a white symmetric Bernoulli signal, it results from the previous discussion that the RVs x k are singular as soon that |a| < 1/2, and we say that the signal x k is singular. Indeed in order to come at (2) it suffices to introduce ŵk = w n-k and it is clear that the RVs ŵk are still IID Bernoulli. The simplicity of this signal explains why it was said previously that singularity is a common phenomenon.

But these results can be extended to a large of other signals. This can be done either by changing the filter that yields x k from w k or by changing the statistical properties of the input.

DRAFT February 22, 2018

Let us first present some preliminary considerations. We restrict our discussion to dynamical discrete time filters. These filters are characterized by the facts that they are causal and that their transfer function H(z) is a rational function of z. Such filters are defined by their poles and their zeros and almost all the filters used in signal processing are dynamical. We consider further only IIF filters, because RIF filters cannot introduce singularity. Indeed this property, as seen previously, is due to a series and the input-output relationships in RIF filters is a simple sum. This means that we exclude from our analysis transfer functions with only one pole at the origin.

We consider also white discrete-valued inputs signals w k , or sequences of IID random variables taking only a finite number q of possible values. Let finally introduce a circle called circle of singularity with center O and with the radius equal to 1/q. This allows to introduce the following theorem.

Theorem 1: Let x k be the output of a dynamical non-RIF filter generated by the input w k . If w k is a white signal taking only q values and if the poles of the filter lie inside the circle of singularity, then the output x k is singular.

Proof: As previously it is made in two steps: 1. Proof that the SM is zero, 2. Proof that there is no discrete component in the DF.

Let F be a dynamical filter defined by its transfer function H(z) or its impulse response h k . Let F be the filter with the impulse response g k = q k h k . It is obvious that its transfer function is G(z) = h(z/q). This implies that if the poles of F are z i , those of F are qz i . The assumption that the poles of F are inside the circle of singularity implies that the poles of F are inside the unit circle, or that F is a dynamical filter. As such a filter is stable we deduce that ∞ k=0 |g k | < ∞. The signal x p is defined by x p = ∞ k=0 h k w p-k and by introducing x p = X and ŵk = w p-k we have X = ∞ k=0 h k ŵk , where the ŵk s have the same properties as the w k s or are IID and take only q values. As indicated above the finite sum X N and the rest R N are defined in (3) where a k is replaced by h k and w k by ŵk .

Let A be the greatest possible value of w k . We have then

|R N | < Aρ N with ρ N = ∞ k=N +1 |h k |.
As X N can take only q N values, the SM S of X satisfies S ≤ 2Aq N , ρ N . This is valid for all N. Then S ≤ 2A lim N →∞ (q N , ρ N ). But we have

q N ρ N = q N ∞ k=N +1 |h k | < ∞ k=N +1 q k |h k | = ∞ k=N +1 |g k |, (8)
and the limit is 0 because the filter F' is stable or ∞ k=0 |g k | < ∞ . This implies that S = 0, or a 1 = 0. The proof that a 2 = 0 is exactly the same as previously and comes only from the whiteness of the w k s or from their independence. Another proof of the fact that there is non discrete component is given by Lukacs (1970, p. 65).

February 22, 2018 DRAFT

It is important to note that this theorem yields only a sufficient condition of singularity. However the condition that the poles lie in the singularity circle is not at all necessary. Examples of this situation can easily be found.

Note is that the possible values of w k do not play any role in the result. However for a given set of values it is sometimes possible to extend the domain of singularity by more abstract methods [START_REF] Peres | Self similar measures and intersections of Cantor sets[END_REF]. But if we impose the singularity for all the possible sets of values, we come back to the conditions of Theorem 1.

The final point is that it is in general no longer possible to interpret the result by a tree of construction. This is especially the case when the dynamical filter has complex poles as for example when the impulse response is of the a k cos(kφ).

IV. THE INFLUENCE OF THE CORRELATION

The assumption of independence, or of whiteness of the input, plays a fundamental role in the previous results. It is introduced in all the papers indicated in the list of references. This assumption allows us to show that, even if the SM is zero, there is no discrete component in the DF, of that a 2 = 0. Thus appears immediately the question of knowing whether it is still possible to meet singularity in the case of colored inputs.

Consider a filter F defined by its impulse response hk and satisfying the conditions of Theorem 1.

The problem of singularity of the output depends on the properties of the RV X given by [START_REF] Hill | Random geometric series and intersymbol interference[END_REF] where the condition h k > 0 is relaxed. We assume that the w k s are symmetric Bernoulli, but not necessarily independent. The maximum value of X is X m = ∞ k=0 |h k | which is finite because F is stable. Let X N be the partial sum analogue to (3) and defined by

X N = N k=0 h k w k . (9)
It takes at the maximum 2 N +1 distinct values v N i and we assume that this maximum is reached. This assumption of distinct values is obviously satisfied when there is no crossing of the branches of the tree constructed with the h k . This appears with filters satisfying [START_REF] Kershner | On symmetric Bernoulli convolutions[END_REF], or for the Cantor-type structure. It is clear that this assumption depends only of the impulse response h k of the filter. It is however satisfied by a large class of filters which is not discussed here. This means that for any N there is no pair (i, j),

i = j such that v N i = v N j .
Because of the symmetry of the w k s, the RVs X and X N are also symmetric and this implies that v N i = 0. Indeed the symmetry and the existence of a zero value would imply that the number of distinct values is odd, which is not the case.

DRAFT February 22, 2018

To each value v N i we associate a node V N i in the tree of construction similar to the one appearing in figure 1. The assumption of distinct values v N i means that the nodes of the tree are single, which means that each node V N i is reached by only one path coming from only one node V N -1 j at the step N -1.

This is obviously satisfied when there is no crossing of the branches of the tree constructed with the h n s. This especially appears if condition ( 6) is satisfied. However, since h k is not necessarily positive as in figure 1, it is not possible to restrict this tree to the nodes V N i corresponding to positive values v N i . Then we assume that there are

2 N +1 distinct nodes satisfying v N 0 < v N 1 < v N 2 .... < v N 2 N +1 -1 .
Finally we assume that the nodes V 0 0 and V 0 1 defined by -v 0 0 = v 0 1 = |h 0 | are generated from an origin node V which does not correspond to a value of X N .

The fundamental consequence of the assumption of distinct values v N i is that for any N and i there is a unique path going from V to v N i . Let i N k (i), 0 ≤ k ≤ N -1, be the indices j characterizing the nodes V k j of this path. These nodes can then be written

V k i N k (i) .
The problem is to calculate the probabilities

p N (i) = P [X N = v N i ], 0 ≤ i ≤ 2 N +1 -1. (10)
When the w k s are IID this probability is 1/2 N +1 . When they are no longer independent, its calculation is much more complicated.

For this we introduce the conditional probability

p N (i, j) = p N [X N = v N i |X N -1 = v N -1 j ]. (11)
called also transition probability. It has two fundamental properties for the discussion that follows.

The first comes from the fact that, as any probability, it is normalized or satisfies for all j the relation

2 N +1 -1 i=0
p(i, j) = 1. However a node V N -1 j of the tree of construction generates only two nodes

V N i
characterized by the indices i + (j) and i -(j) and, according to [START_REF] Naraghi-Pour | On the continuity of the stationary state distribution of DPCM[END_REF], corresponding to the values

v N i± v N -1 j ± h N w N .
As a consequence for a given j there is only two terms in the previous sum and we have

p N [i + (j), j] + p N [i -(j), j] = 1 (12)
The second starts from the fact that there is only one V N -1 j at the step N -1 of the tree generating V N j and called j(i) Thus p N (i, j) is zero except when j = j(i), and the only non-zero values of p N (i, j) are

q N (i) = p N [i, j(i)] (13)
for N > 0 and q 0 (i) = 1/2.

February 22, 2018 DRAFT

It results from [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF] and from the unicity of the path between V and

V N i that p N [(X N = v N i ).(X N -1 = v N -1 j )] = p N -1 (j).p N (i, j)δ[j -j(i)]. (14)
where δ[.] is the Kronecker delta symbol. By a summation on j, which contains only one term, we obtain

p N (i) = p N -1 [j(i)]q N (i). (15)
By repeating this at all the nodes of the unique path between V and V N i characterized by the indices v N k (i) we obtain

p N (i) = N k=0 q k [i N k (i)]. (16)
When the RVs w k s are IID we have of course q k [i N k (i) = 1/2, and we find again that the values v N i have equal probabilities 1/2 N +1 .

The probabilities p N (i) of ( 16) are normalized, or i p N (i) = 1, where the sum is extended to all the indices i from 0 to 2 N -1 . This property is valid for N = 0 because q N (i) = 1/2. Suppose that it is valid at the step N -1. Since each node V N -1 l generates only two nodes the result comes from [START_REF] Peres | Self similar measures and intersections of Cantor sets[END_REF].

The relation ( 16) is the basis for the discussion of the singularity. Indeed if all the p N (i) tend to 0 when N → ∞ there is no value v ∞ i with a finite probability, and this means that the RV cannot have a discrete component and then is singular. This can be specified by the following theorem.

Theorem 2: Let X be the RV ∞ k=0 h k w k , where h k is the impulse response of a dynamical non-RIF filter F and w k a sequence of Bernoulli RVs. If the poles of F are inside the circle of singularity, if the possible values v N i of the partial sums X N are distinct and if the transition probabilities q N (i) defined by (13) satisfy

0 < q N (i) < B < 1, (17)
then the RV X is singular, or a 1 = a 2 = 0.

Proof: If the poles are inside the circle of singularity, Theorem 1 shows that the SM is zero or a 1 = 0.

It remains to show that a 2 = 0. This is a direct consequence of ( 16) and ( 17) because p N (i) < (1/2)B N , which tends to zero when N → 0.

Comments: It is clear that this situation appears for white input because in this case q N (i) = 1/2. The question that remains concerns the conditions of the theorem on the filter. It is clear from the previous discussion and that if h k = a k with a < 1/2, these conditions are satisfied. There is a large class of filters satisfying also these conditions. However the question of characterizing all the dynamical filters with poles inside the circle of singularity and introducing distinct values v N i remains open. As a matter of fact it is possible to extend this theorem to the case where these values are not distinct, but this introduces other conditions that cannot be presented in this paper.

DRAFT

February 22, 2018

V. CONCLUSION When the input of a causal discrete-time exponential linear filter with impulse response a k is a Bernoulli white noise, the output is singular for a < 1/2. A simple proof of this result was presented and it exhibits two steps. The condition on the parameter a and the fact that the input is a discrete-valued signal implies that the SM of the output is zero, which means that there is no continuous component in its DF. The fact that this DF does not contain a discrete component arises from the whiteness of the input. This result is not specific to exponential filters. We have shown that singularity can appear in many other situations and we have established a sufficient condition for singularity by using the positions of the poles of the filter with respect to the circle of singularity and the whiteness of the input.

This assumption of whiteness can however be partially relaxed without changing the singularity of the output. Some sufficient conditions ensuring the singularity with colored inputs have been established.

These conditions are obviously satisfied not only by white noise but also by a large class of correlated signals. It is especially the case of some Markovian signals of finite order. The theoretical analysis also

shows that the output generated by colored inputs can be a mixture of a discrete and a singular distribution.

Computer experiments in order to verify the theoretical results will be discussed in a forthcoming paper.

  Finally mixtures of discrete and and singular random signals can be obtained depending on the properties of the correlation function of the input, and theoretical and experimental examples of such signals are presented. DRAFT February 22, 2018

  X is in general continuous, but can also be singular for values of a belonging to

	a set of zero measure.
	Point 2 can be shown directly by a simple calculation. Point 3 is without interest for this paper and the
	complete set of points of singularity is still a subject of research. Point 1 was shown long time ago in the
	framework of infinite products of convolutions (Kershner and Wintner 1935) and discussed more recently
	(Peres and Solomyak 1998, Solomyak et al. 2000). Because of its importance for all what follows, we
	shall now present a direct proof.

February 22, 2018 DRAFT


S Benhamadouche
email: sofiane.benhamadouche@edf.fr

On the use of (U)RANS and LES approaches for turbulent incompressible single phase flows in nuclear engineering applications

Keywords: LES, (U)RANS, Nuclear engineering, Wall-resolved, Wall-modeled, HPC

The present paper gives some ideas and guidelines one should keep in mind while running (U)RANS or LES computations. The paper starts with (U)RANS approaches, advocating the use of Reynolds Stress Models for complex flows and recommending further work on modeling of turbulent heat fluxes, which remains today too basic in industry. The superiority of wallresolved models vs. wall-modeled in RANS is recalled and the use of adaptive wall treatment is suggested. The concept of Unsteady RANS is finally questioned. Then, important issues around LES are raised. The mesh refinement criteria are recalled for wall-resolved LES and the use of wall models addressed. The production of DNS and wallresolved LES calculations for flow understanding and RANS validation is encouraged.

Introduction

(Unsteady) Reynolds Averaged Navier-Stokes ((U)RANS) and Large Eddy Simulation (LES) approaches are today widely used in nuclear engineering applications for predicting the dynamics and heat transfer of turbulent flows. Both approaches exist since a long time and will remain useful for understanding several industrial configurations for several decades, playing different roles. The present overview is restricted, for the sake of clarity, to (U)RANS and LES for configurations in which the flow is incompressible and single phase as we consider that a lot of work is still required for this type of flow and for these two approaches. This article is in the continuity of an invited lecture by [START_REF] Chabard | Heat and fluid flow simulations for deciding tomorrow's energy[END_REF] at Turbulence Heat and Mass Transfer conference 6. In this contribution, the authors insisted on the importance of RANS approaches in general and in particular on Reynolds Stress Models (RSMs) without forgetting the role of LES. Detached Eddy Simulation (DES), Scale-Adaptive Simulation (SAS) ... turbulence modeling approaches are out the scope of the present article. Verification and Validation (V&V), which is a necessary procedure to check the correct numerical solution of the equations and the ability of a model to predict standard flows is a prerequisite but will not be addressed. Neither can Uncertainty Quantification (UQ) but the hundreds of simulations required for the latter make fine LES unaffordable, incompatible today, with UQ. The statements given in the following largely assume the use a standard finite volume approach as this is the most common discretization used in nuclear engineering. Regarding heat transfer, nothing particular will be said for very low Prandtl numbers and one assumes values around 1 which is the case in Pressure Water Reactors (PWR). Finally, as turbulence modeling is full of acronyms, the author does his best to reduce them and to recall them, at least for the first use.

On the one hand, steady and unsteady RANS turbulence models are frequently used in nuclear engineering applications as they are more affordable than finer approaches. While looking at the literature, it is obvious that the users are more inclined to use Linear Eddy Viscosity Models (LEVMs), especially stable and widely known/recommended ones such as the Realizable k-ε, the k-ω SST ... This is legitimate, but are they the best models to be used in nuclear engineering configurations? Why our use of RSMs is still so marginal? Wall-resolved models, often called Low Reynolds Number models (LRN) exist for both Eddy Viscosity Models (EVMs) and RSMs. These models are still rarely used due to their cost. Is it possible today to avoid the use of wall functions/boundary layer modeling, at least in some applications? While performing heat transfer computations, as several computations are done using LEVMs, almost all the contributions use a Simple Gradient Diffusion Hypothesis (SGDH) which utilizes the concept of the turbulent Prandtl number. Limitations of such assumptions will be shown later. Is the concept of wall functions/modeling really avoidable, in particular for large computational domains or very high Reynolds numbers? One can find several contributions using an Unsteady RANS when LES is difficult to carry out or unaffordable and when quantifying the unsteadiness is needed such as in Fluid Structure Interaction or thermal fatigue issues. Is the spectral content of a URANS computation always meaningful?

On the other hand, LES is becoming more and more utilized thanks to the growth of High Performance Computing (HPC) resources (speed, memory, storage). The present article focuses on standard use of LES using sub-grid scale eddy viscosity concept as it is the one widely used in industry. What are the space discretization criteria that make us confident in a LES simulation? Due to standard limitations in computational resources, wall-modeled LES using wall functions or even hybrid RANS/LES approaches are still the solution to tackle high Reynolds numbers. Is this approach realistic? Finally, few words are said about HPC projections in the near future as this is one of the major evolutions that one may expect, in particular for LES.

(U)RANS approaches 2.1. Regarding the use of LEVMs vs. RSMs

Linear Eddy Viscosity Models might be very useful, even combined with wall functions, in order to obtain reliable global quantities such as pressure drops, mass flow rates, averaged Nusselt numbers ... Many examples exist in the literature showing that these approaches are sufficient to reasonably predict these kinds of quantities, or even more local ones, such as mean velocity or mean temperature profiles. [START_REF] Wells | Overview of CFD Round Robin Benchmark of the High Fidelity Fuel Rod Bundle NESTOR Experimental Data[END_REF] give an overview of a benchmark organized by EPRI where several models have been used to predict the flow through a fuel assembly. LEVMs seem sufficient to predict the pressure drop through a grid with mixing vanes (see Conner et al., 2015 who show that the Realizable k-ε gives a good estimate of the pressure drop through a mixing vane grid). However, [START_REF] Benhamadouche | Pressure drop predictions using Code_Saturne in NESTOR CFD Benchmark[END_REF] shows that the utilized RSM (SSG model, see [START_REF] Speziale | Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach[END_REF], on the same configuration, is very sensitive to the discretization scheme and produces good results with an accurate numerical scheme. Thus, the RSMs are much more sensitive to the discretization errors than the EVMs and should be used with special care. One can find in the literature conclusions such as "The Reynolds Stress Model gives poor results compared to the k-ε"! As the latter is just the trace of the first one, this is not intellectually speaking satisfactory. Something must be wrong with the use of the model (numerical approach, inlet boundary conditions ...). Furthermore, many of those using RSMs in the literature rarely give details about the specific model used. One may find sentences such as "The RSM has been tested ...". On the contrary, when they use an EVM, the precise version of the model is given; standard k-ε [START_REF] Launder | The numerical computation of turbulent flows[END_REF], Realizable k-ε [START_REF] Shih | A New k-ε Eddy-Viscosity Model for High Reynolds Number Turbulent Flows -Model Development and Validation[END_REF], k-ω SST [START_REF] Menter | Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications[END_REF] ... There are several RSMs in the literature; among them the LRR [START_REF] Launder | Progress in the Development of a Reynolds-Stress Turbulence Closure[END_REF] and the SSG [START_REF] Speziale | Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach[END_REF], with several variants that may affect the constants or some terms in the Reynolds stresses and dissipation equations (see [START_REF] Pope | Turbulent flows[END_REF]. There are also numerous test cases and publications and even a whole book (Hanjalić and B. Launder, 2011) demonstrating the necessity of resorting to RSMs when body forces (buoyancy, rotation, swirl) induce specific anisotropy to the stress which is then no longer aligned on the mean velocity gradient. These remarks show that, on the one hand, RSMs (or the use of RSMs) are probably not mature enough in an industrial context and, on the other hand, they need to be used more often in order to obtain a wealth of feedback similar to the one existing for EVMs. [START_REF] Bellet | Swirling and secondary flows in PWR primary loops, CFD might bring some light[END_REF] focused on the importance of turbulence modeling to predict swirling flows. The case shown in Fig. 1 clearly emphasizes the inability of EVMs, even advanced ones such as the v²f (here the ϕ= v²/f -model introduced by [START_REF] Laurence | A robust formulation of the v2-f model[END_REF], to predict the circumferential flow in a vortex tube. The test-case is taken from [START_REF] Escudier | Observations and LDA measurements of confined turbulent vortex flow[END_REF] (see Fig. 1 top). The swirl is globally created in the large cylinder then goes inside the small one. Even at high small to large tube diameter ratios, EVMs are unable to correctly predict the flow structure (see Fig. 1 bottom). However, the Reynolds Stress Model (SSG model here with Scalable Wall Functions, see [START_REF] Esch | Heat transfer predictions based on two-equation turbulence models with advanced wall treatment[END_REF] and the LES predict the flow rather well. Note that for small diameter ratios, only LES is able to predict the vortex breakdown phenomenon which appears. The ability of LES to predict such flows has been already shown by [START_REF] Derksen | Simulations of confined turbulent vortex flow[END_REF] and is confirmed here. This kind of example shows that when rotating or secondary flows play a major role, such as the flow through mixing grids or in the hot legs of a PWR upper plenum, a second moment closure (RSM) should be used by default in order to capture the anisotropy of turbulence and, hence, obtain satisfactory local quantities. A last argument used sometimes to justify the use of an EVM instead of a RSM is the cost of this latter. As for incompressible solvers, the pressure correction step in much more expensive than solving transport equations, this is in the author's opinion an argument out of date.

Fig. 1.

Top: confined Vortex Tube Geometry, Re(inlet)=1900, ratio of angular to axial momentum Ω=4.14, bottom: mean axial and tangential velocity in the large pipe (198.5 mm from the closed end).

Wall-resolved vs. wall-modeled RANS approaches

Utilizing a wall-resolved RANS model (often called in specialized literature Low Reynolds Number (LRN) models) is becoming more and more affordable when the Reynolds number is moderate to high (depending on the definition of the Reynolds number, this gives Reynolds numbers going from 10 4 to 10 7 ). This kind of models is obviously better than the one using a wall-modeled approach at the wall. If one uses the same wall-resolved RANS model (such as the k-ω SST from [START_REF] Menter | Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications[END_REF], the k-ε from [START_REF] Launder | Application of the energy dissipation model of turbulence to the calculation of near a spinning disc[END_REF], the v²f from [START_REF] Durbin | Near-wall turbulence closure modeling without damping functions[END_REF] or one of its more robust versions, the Elliptic Blending Reynolds Stress Model from [START_REF] Manceau | Elliptic blending model: A new near-wall Reynolds-stress turbulence closure[END_REF]), as a basis, one can't expect obtaining better results with a wall model either with an algebraic approach (so called "wall functions" approach, see [START_REF] Craft | Progress in the generalization of wall functions treatment[END_REF] or even with a numerical one (solving 1D differential equations in the near wall cell, see [START_REF] Craft | A new wall function strategy for complex turbulent flows[END_REF]. Thus, when it is possible, the calculation should be done with a wall-resolved model, fulfilling the well-known requirement y+<1 everywhere. Here, y+ is the nondimensional distance to the wall of the first computational cell based on the friction velocity and the kinematic viscosity. Of course, a grid refinement study has to be carried out in order to obtain a numerically converged result.

However, even if the case can be, theoretically speaking, computed using a wall-resolved RANS model, the complexity or the size of the geometry may lead, depending on the meshing strategy, to regions where it is impossible to refine and thus which need a wall treatment. Solutions to mix wall-resolved and wall-modeled-regions, called "adaptive wall treatment" in a single mesh have been developed by [START_REF] Popovac | Compound wall treatment for RANS computation of complex turbulent flows and heat transfer[END_REF] and [START_REF] Billard | Adaptive Wall Functions for an Elliptic Blending Eddy Viscosity Model Applicable to Any Mesh Topology[END_REF] for EVMs and by [START_REF] Wald | Adaptive wall treatment for the Elliptic Blending Reynolds Stress Model[END_REF] for RSMs. A mesh cut downstream a mixing grid in a fuel assembly is shown in Fig. 2. The meshing strategy uses here a block-structured method with ICEM CFD. One can see the traces of the dimples and springs and the variations in the refinement obtained around a rod. This shows the importance of developing adaptive wall treatments, either with wall functions or a 1D numerical approaches, departing from a wallresolved model.

Wall-modeled approaches might give satisfactory results in some configurations, in particular for predicting global quantities. This is not always true, that is why a grid sensitivity study is highly recommended whenever one uses this kind of approaches. An example is given in Fig. 3. Several computations (not all shown here but the reader can refer to Benhamadouche et al., 2015aBenhamadouche et al., , 2015b) ) have been performed for the flow through a square edged orifice plate at a Reynolds number equal to 25000, based on the bulk velocity Ub in the main pipe and the diameter of this latter. The ratio of the orifice diameter to the pipe diameter is = 0.62, and the ratio of the orifice thickness to the pipe diameter is 0.11. The target value is the discharge coefficient through the contraction. Fig. 3 shows the discharge coefficient measured 1D upstream of the orifice and 0.5D downstream (both measured from the upstream face of the contraction as required in ISO 5167-2, 2013). EVMs, not shown here, exhibit similar mesh sensitivity as the wall-modeled RSM utilized here (the SSG of Speziale et al., 1991 combined to the standard two velocity scales wall function). The ISO correlation estimate obtained from ISO 5167-2 (2013) represented in the figure gives the target value (however, only the behavior of the numerical results is important here). The wall-resolved RANS approach uses the Elliptic Blending (EB) concept introduced by [START_REF] Manceau | Elliptic blending model: A new near-wall Reynolds-stress turbulence closure[END_REF] with SSG Reynolds Stress Model far from the walls (note that the same concept exists for EVMs, see [START_REF] Billard | A robust k-ε-v²/k elliptic blending turbulence model applied to near-wall separated and buoyant flows[END_REF]. This model is inspired from elliptic relaxation introduced by Durbin (1993) but using only one elliptic equation to model the wall damping instead of six, one for each Reynolds stress as initially suggested. This wall-resolved RANS model is called EB-RSM for Elliptic Blending-Reynolds Stress Model. Fig. 3 shows the discharge coefficient across the orifice as a function of the refinement. Note here that the characteristic non dimensional mesh size in the abscissa is representative of the wall normal one. When the mesh is refined in the wall normal direction, it is also refined in the stream-wise and azimuthal directions using the same ratios. Two different meshes are used for the wall-modeled approach (see Benhamadouche et al., 2015b) For the coarsest refinement of Mesh2, the non-dimensional wall distance y+ is approximately 55 upstream of the orifice and increases up to y+≈150 where the flow reattaches. For the finest refinement of Mesh2, y+≈22 upstream of the orifice and y+≈60 where the flow reattaches. The non-dimensional wall distances for the refinements of Mesh1 lie within the range of those of Mesh2. The corresponding values for the coarsest mesh of the wall-resolved model are y+≈1 upstream of the orifice and y+≈2 where the flow reattaches. The refinement ratio between two successive meshes is 1.3 and is applied equally in the streamwise, radial and azimuthal directions. As one can notice, the wall-resolved model results are almost grid independent, whereas wall-modeled approach based on the same homogeneous model (SSG model far from the walls), exhibits a strong sensitivity to the mesh refinement. In the present case. Only the wall-modeled approach is able to predict a discharge coefficient not very sensitive to the mesh refinement and within less that 1% of the ISO correlation value (a wall-resolved LES in Benhamadouche et al. (2015a) gives a discharge coefficient very close to the one given by the ISO).

This example shows the systematic and logical superiority of wall-resolved or Low Reynolds Numbers approaches compared to wall modeled ones (algebraic or numerical wall treatment). However, wall-resolved approaches will probably remain impossible for decades to use systematically either for very large computational domains (a whole nuclear vessel for example) or for very large Reynolds numbers (such as in the hot legs of a 1300 MWe where the Reynolds number is of the order of 10 8 ). In these cases, the use of specific near wall treatments, such as the "adaptive wall treatment" suggested above, will remain mandatory for many years.

Heat transfer predictions with RANS approaches 2.3.1. With Eddy Viscosity Models

As a majority of industrial contributions still use EVMs and particularly, LEVMs, one starts by these models. It is impressive to see the number of users who still utilize the Simple Gradient Diffusion Hypothesis (SGDH) approach even with buoyancy effects. It is possible with an EVM to increase models' capabilities by introducing more sophisticated models (see [START_REF] Kenjereš | Contribution to elliptic relaxation modelling of turbulent natural and mixed convection[END_REF] such as the Generalized Gradient Diffusion Hypothesis (GGDH), the AFM (Algebraic Flux Model) which can take into account buoyancy effects, or even more advanced models based on transport equations for the turbulent heat fluxes. The little interest for modeling the turbulent heat fluxes has several reasons. First, the lack of experimental or Direct Numerical Simulation data compared to the ones available for the dynamics. The lack of experimental data can be explained by the difficulties one may face to carry out such experiments and the inherent financial cost. Some valuable DNS data have been made available for turbulent flows with heat transfer (see [START_REF] Kasagi | Direct numerical simulation of combined forced and natural turbulent convection in a vertical plane channel[END_REF][START_REF] Kasagi | Direct numerical simulation of combined forced and natural turbulent convection in a vertical plane channel[END_REF][START_REF] Trias | Direct numerical simulations of two and three-dimensional turbulent natural convection flows a differentially heated cavity of aspect ratio 4[END_REF] among others) but much more efforts are still needed in particular for conjugate heat transfer cases. For the latter, DNS data is rare, with the exception of the work done by [START_REF] Tiselj | Double-sided cooling of heated slab: Conjugate heat transfer DNS[END_REF] and the ongoing work by [START_REF] Flageul | DNS of turbulent channel flow with conjugate heat transfer: Effect of thermal boundary conditions on the second moments and budgets[END_REF], whereas with current DNS codes and resources, the scientific community could easily be more prolific in this area.

The next example is often studied as coolant flows in the cores of current gas-cooled nuclear reactors which contain ascending vertical flows in a large number of parallel passages (see [START_REF] Keshmiri | Turbulence Models and Large Eddy Simulations Applied to Ascending Mixed Convection Flows[END_REF]. The flow going upwards in a vertical heated pipe is the analogous academic test case. The flow properties are considered constant and the Boussinesq approximation is used. While increasing buoyancy, the flow experiences a re-laminarization. This transition, which may occur in other situations and which dramatically affects heat transfer, may be found in other industrial configurations. Both [START_REF] Kim | Assessment by comparison with DNS data of turbulence models used in simulations of mixed convection[END_REF] and [START_REF] Keshmiri | Turbulence Models and Large Eddy Simulations Applied to Ascending Mixed Convection Flows[END_REF] studied this flow with EVMs and Fig. 4 gives a fair illustration of the results. They concluded that several EVMs fail in predicting the mixed convection regime. One may notice in particular that the widely used k-ω SST exhibits good results for low and high buoyancy parameter values but totally fails in predicting the transitional regime. Note that the computations have been carried out with two different codes (see Fig. 4). More recent models such as the ones based on elliptic relaxation concepts tend to better predict such flows as they include wall normal velocity mean fluctuations and turbulent length-scale characteristics (ϕ=v²/f model results in [START_REF] Keshmiri | Turbulence Models and Large Eddy Simulations Applied to Ascending Mixed Convection Flows[END_REF]. Introducing more physics in the EVMs seems then necessary to better predict heat transfer phenomena.

Going beyond Eddy Viscosity Models

Another way to enrich the physics is to use a RSM combined with an advanced treatment for the turbulent heat fluxes using at least a GGDH or an AFM or if possible a Differential Flux Model (DFM, see [START_REF] Dol | A DNS-based thermal second moment closure for buoyant convection at vertical walls[END_REF] among others) in which transport equations are solved for the turbulent fluxes. Several authors already introduced these approaches but these are still far from mainstream. Recently, [START_REF] Dehoux | Algebraic modelling of the turbulent heat fluxes using the elliptic blending approach -application to forced and mixed convection regimes[END_REF] and Dehoux (2012) used the GGDH, AFM and DFM in conjunction with the EB-RSM (Elliptic Blending-Reynolds Stress Model) and extended the elliptic blending concept for modeling the turbulent heat fluxes. Fig. 5 shows the numerical results obtained with different approaches for the temperature variance in a turbulent channel flow in the forced and mixed convection regimes. These profiles show a progressive enhancement of the temperature variance as the turbulent heat flux model is enriched (from SGDH to DFM, then, from EB-GGDH to EB-DFM, see [START_REF] Dehoux | Modélisation statistique des écoulements turbulents en convection forcée, mixte et naturelle[END_REF]). Note that the model here used for the dynamics is the EB-RSM. One may also notice the poor behavior of the two advanced EVMs (kω SST and ϕ-model) in the mixed convection regime.

Other more complex flows, not shown here, such as the one in a heated cavity and a PWR upper plenum and its hot legs confirm the need of advanced models for the turbulent heat fluxes, at least algebraic ones. 2004). Right: mixed convection, channel flow, Reτ=150, Gr=9.6 10 5 , Pr=0.71, comparisons to DNS data from [START_REF] Kasagi | Direct numerical simulation of combined forced and natural turbulent convection in a vertical plane channel[END_REF].

On Unsteady RANS

RANS equations in their unsteady form (so called URANS) are widely used when unsteadiness is needed. Their use for example in fuel bundle configurations is widespread in particular while aiming at predicting gap instabilities (see [START_REF] Chang | Simulations of turbulence, heat transfer and mixing across narrow gaps between rod-bundle subchannels[END_REF][START_REF] Chang | Simulations of turbulence, heat transfer and mixing across narrow gaps between rod-bundle subchannels[END_REF][START_REF] Meyer | From discovery to recognition of periodic large scale vortices in rod bundles as source of natural mixing between subchannels-a review[END_REF]. The approach is attractive since DNS or even LES are still impossible to carry out with standard computational resources at the core Reynolds numbers (see [START_REF] Tavoularis | Reprint of: Rod bundle vortex networks, gap vortex streets, and gap instability: A nomenclature and some comments on available methodologies[END_REF]. First, as it has been reported by [START_REF] Tavoularis | Reprint of: Rod bundle vortex networks, gap vortex streets, and gap instability: A nomenclature and some comments on available methodologies[END_REF], one should be very cautious about the interpretation of the regular coherent structures predicted in URANS. The question raised hereafter is about the meaning of unsteadiness with URANS. The following example shows that if one observes unsteadiness with URANS on a given mesh, it might disappear while refining the mesh and in this case what started out as URANS ends up as a steady state RANS. Of course, the instability of the following case has nothing to do with gap instability but shows that a popular case which has been considered for a long time as a URANS case, in particular with RSMs, is actually a RANS one and no unsteadiness should be expected. [START_REF] Fadai-Ghotbi | Revisiting URANS computations of the backwardfacing step flow using second moment closures. Influence of the numerics[END_REF] computed the 2D backward facing step flow and proved that the unsteadiness of the vortex shedding decreases while refining the mesh by using a wall-resolved RSM. They also showed that 3D computations give a steady converged solution. Fig. 6 shows the maximum value of the ratio resolved energy/total energy as a function of the stream-wise location while progressively refining the mesh (from mesh 7 to 11, mesh 12 giving a steady solution). This example clearly campaigns for verifying the evolution of the resolved energy while performing mesh sensitivity/convergence studies.

LES approaches 3.1. Quasi-DNS and wall-resolved LES

More and more wall-resolved LES are found in the literature (see for example [START_REF] Afgan | Large eddy simulation of the flow around single and two side-by-side cylinders at subcritical Reynolds numbers[END_REF]. These computations are sometimes called Quasi-DNS [START_REF] Komen | Quasi-DNS capabilities of OpenFOAM for different mesh types[END_REF] or even DNS [START_REF] Baglietto | CFD and DNS methodologies development for fuel bundle simulations[END_REF] providing valuable data, such as Reynolds Stresses budgets or turbulent heat fluxes (see [START_REF] Shams | Quasi-direct numerical simulation of a pebble bed configuration, Part-II: Temperature field analysis[END_REF], for RANS modeling. However, in the author's opinion, the nomenclature is not adequate as in numerous examples the discretization is only second order in space or less as sometimes a bounded central differencing scheme is introduced for the convection term. Rhie and Chow interpolation in the pressure correction step is utilized with the collocated arrangement and introduces additional numerical errors. These computations should rather be called wall-resolved LES or very fine LES. With the growth of computing power and thanks to advanced numerical techniques, proper DNS computations for complex flows, higher Reynolds numbers and even more complex geometries are today available (see for example [START_REF] Ranjan | Direct simulation of turbulent heat transfer in swept flow over a wire in a channel[END_REF]Dairay et al., 2015, among others) and there is a hope that this will continue in the future.

Wall resolved LES space refinement requirements

Still several communications dealing with wall-resolved LES insist on the refinement in the wall normal direction (∆y+<1) but not on the stream-wise and span-wise ones. It is commonly admitted today that for a turbulent channel flow and using a second order discretization scheme, one has to respect the two additional criteria; ∆x+<40 and ∆z+<20, where x and z are the streamwise and span-wise coordinates, respectively. Fig. 7 gives an illustration of the results one may obtain while only refining in the wall normal direction without taking care of the other directions (here ∆x+ and ∆z+ go from 100 to 1000). With this computation, the predicted shear stress is too important due to artificial that appear at the wall. The computation using wall functions (with the same refinement in x and z directions) but a first computational point that satisfies 30<∆y+<100 gives better results.

However, the wall refinement is not the only criteria that should prevail. One has ideally to check whether the computational cells are all in the inertial sub-range, let's say of the order of the Taylor-micro length scale. One may perform a precursor RANS computation which gives of course reasonable results. This computation can give a reasonable estimate of the Taylor micro-scale and the LES mesh can then be created (one should take care of locations where the flow exhibits too low turbulent Reynolds numbers). The example of an estimate of the Taylor micro-scale is given in Fig. 8 for the flow along a ribbed wall. The LES mesh has been created based on this scale and the computation gave very satisfactory results (see [START_REF] Keshmiri | Assessment of Advanced RANS Models against Large Eddy Simulation and Experimental Data in the Investigation of Ribbed Passages with Passive Heat Transfer[END_REF].

A posteriori checks should also be performed in order to check whether the turbulent kinetic energy is reasonably resolved. [START_REF] Afgan | Large eddy simulation of the flow around single and two side-by-side cylinders at subcritical Reynolds numbers[END_REF] showed this kind of a posteriori estimations by computing an estimate of the sub-grid scale dissipation rate and turbulent kinetic energy (see Fig. 9). This estimate is much better and has more sense than using the ratio of the sub-grid scale viscosity to the kinematic viscosity. If the estimated sub-grid scale turbulent kinetic energy is everywhere lower than 5% of the total turbulent kinetic energy, one can consider that the Large Eddy Simulation is fine enough. Estimates of the sub-grid scale turbulent kinetic energy ( 2

3 3 / 2       ∆ = π ε SGS K SGS c k with j i j i t SGS x u x u ∂ ∂ ∂ ∂ = ' ' ν ε ).

Wall modeling in LES?

As it has been shown in Fig. 7, wall-modeled LES might give correct mean values. However, its use is still subject to discussion and objections (see [START_REF] Piomelli | Wall-Layer Models for Large-Eddy Simulations[END_REF]. This approach, which often uses wall functions concepts, is a very first step of hybrid RANS/LES technique (using a logarithmic profile for example). the complexity of the flow dynamics in the near wall region, there is rather weak correlation between an instantaneous velocity at a given distance from the wall and the instantaneous shear-stress under it and only long time averaged values are well rendered, by construction. This approach is thus a priori not recommended for unsteady problems such as thermal fatigue or Fluid Structure Interaction. However, in order to deal with realistic Reynolds numbers, one can test this approach and check whether the low frequencies predicted at the first computational cell are sufficient for the studied problem. Fig. 10 gives an example taken from [START_REF] Benhamadouche | CFD estimation of the flow-induced vibrations of a fuel rod downstream a mixing grid[END_REF] in which the objective is to compute the pressure load along a rod bundle in a typical fuel assembly at Re=40000, based on the hydraulic diameter and the bulk velocity. The pressure load has been used combined to a beam equation to predict the displacements obtained along the central rod and the qualitative results correspond to what is observed in reality (few microns). However, this computation can't be considered as validated and the methodology has to be confronted to experimental data (or why not DNS data with the available computing power today). Fig. 10. Flow through a mixing grid at Re=40000. Left: the computational domain, middle: cut of the computational domain (2x2 periodic bundle), right: snapshot of the instantaneous crossstream velocity.

Towards very large computations?

Several CFD codes are showing very efficient scalability on very large computers, among them: Code_Saturne open source (code-saturne.org) EDF CFD tool (see [START_REF] Archambeau | Code_Saturne: A finite volume method for the computation of turbulent incompressible flows: Industrial applications[END_REF][START_REF] Fournier | Optimizing Code_Saturne computations on Petascale systems[END_REF]. This code is based on a collocated finite volume approach. Fig. 11 gives a tentative road-map of the computation sizes that would be used (this is valid for both RANS and LES approaches). One should first distinguish between standard computations (or "production computations") and "Big" computations (could be named "Research computations"). Of course, the objective of these two kinds of simulations is not the same. Standard computations are today around 10 million cells for a duration of a couple of weeks. Some computations reach the order of 100 million cells in nuclear engineering but are still rare (simulating the whole nuclear vessel for example). The factor that can be gained for this kind of computations in the next five years is not large. A factor of 10 would be very good, considering that standard users will have standard HPC machines with a limited access to the cores and a limited storage memory. It is possible to enhance this projection by enhancing pre and post-processing tools and access to supercomputers. "Big computations" will probably continue to grow to reach up to few hundred billion cells or even with durations of few months. Those who carry out these computations have access to exceptional resources and are rather performing computations using LES or DNS on relatively simple geometries. One should consider also the physical time needed. In the example shown in Fig. 10, the physical time needed to obtain an exploitable pressure load led to run the computation over almost one year on a BlueGene/P with 1024 cores (much more time was needed to predict the pressure load than the statistics). Other physics such as conjugate heat transfer may dramatically increase the physical time needed. In addition, one of the objectives is to increase the Reynolds number. Let's take the example of DNS in a homogeneous isotropic turbulence. If the time step is of the order of the Kolmogorov time scale, increasing the turbulent Reynolds number by a factor of 100 will reduced the time step by a factor of 10 which will make the computation naturally longer. This effect should be less dramatic in LES but there will be also an increase of the needed computational time.

Conclusions and perspectives

The present paper gives some ideas and guidance one should keep in mind while running (U)RANS or LES computations. Here are the major ones concerning both approaches:

-LEVMs can produce satisfactory results in particular while dealing with global quantities, -If the flow exhibits anisotropic turbulence due to rotating or secondary flows, one should rather use a RSM, in particular to obtain local quantities, -Wall-resolved RANS simulations are clearly superior to wall-modeled ones, -Predicted unsteadiness from (U)RANS simulations should be treated with caution as they might be highly dependent on numerical parameters., -Wall refinement is not the only important issue for a wall-resolved LES, the cell sizes far from the wall should be small enough to be in the inertial zone and some a posteriori tests can be performed to check the quality of the computation, Some perspective can be also drawn:

-RSMs and advanced models for turbulent heat fluxes still need much more development, feedback and dissemination, -Adaptive wall treatment for RANS in order to deal with very large Reynolds numbers can replace more standard approaches, -More DNS and very fine LES should be carried out in the future, in particular on complex geometries, to contribute fluid flow understanding and to (U)RANS validation, -Wall modeling in LES will remain questionable, -The physical time needed for DNS and LES will increase in the future leading to longer computations, -More efforts (pre-post processing, access to bigger resources) need to be made to increase the size of "standard" computations.

Acknowledgements

The author would like to acknowledge all the contributors to the numerical simulations shown here. He also warmly acknowledges NURETH-16 organizers who invited the author to give an invited lecture summarized in the present article.

Fig. 2 .

 2 Fig. 2. Mesh cut downstream a grid with mixing vanes.

Fig. 3 .

 3 Fig. 3. Left: Computational domain for the square edged circular orifice plate, right: discharge coefficient across the orifice versus mesh refinement.

Fig. 4 .

 4 Fig. 4. Ratio of mixed-to-forced convection Nusselt number as a function of the buoyancy parameter by several RANS models, LES and DNS for an ascending mixed convection flow (from Keshmiri et al., 2012).

Fig. 5 .

 5 Fig. 5. Temperature variance. Left: forced channel flow, Reτ=640, Pr=0.71, comparisons to DNS data from Abe et al. (2004). Right: mixed convection, channel flow, Reτ=150, Gr=9.6 10 5 , Pr=0.71, comparisons to DNS data from[START_REF] Kasagi | Direct numerical simulation of combined forced and natural turbulent convection in a vertical plane channel[END_REF].

Fig. 6 .

 6 Fig. 6. Maximum resolved energy/total energy ratio a function of the stream-wise location for successive mesh refinements. Mesh 12, finest, gives a steady solution. Image taken from Fadai-Ghotbi et al. (2008).

Fig. 7 .

 7 Fig. 7. Mean stream-wise velocity component downstream a T-junction.

Fig. 8 .

 8 Fig. 8. Flow along a ribbed wall. Re=30000 with P/k=9, H/k=10, b/k=1. The Taylor scale predicted by the ϕ-model (Laurence et al., 2004).

Fig. 9 .

 9 Fig. 9. The flow around two side-by-side cylinders at subcritical Reynolds numbers (Re=3000).

Fig. 11 .

 11 Fig. 11. Tentative roadmap of HPC computations using Code_Saturne.