Antoine Marnat
email: marnat@math.unistra.fr

About Jarník's-type relation in higher dimension

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Introduction

Throughout this paper, the integer n ≥ 1 denotes the dimension of the ambient space, θ = (θ 1 , . . . , θ n ) denotes an n-tuple of real numbers such that 1, θ 1 , . . . , θ n are Q-linearly independent.

Let d be an integer with 0 ≤ d ≤ n -1. We define the exponent ω d (θ) (resp. the uniform exponent ωd (θ)) as the supremum of the real numbers ω for which there exist rational affine subspaces L ⊂ R n such that dim(L) = d , H(L) ≤ H and H(L)d(θ, L) ≤ H -ω for arbitrarily large real numbers H (resp. for every sufficiently large real number H). Here H(L) denotes the height of L (see [START_REF] Schmidt | On heights of algebraic subspaces and diophantine approximations[END_REF] for more details), and d(θ, L) = min P ∈L d(θ, P ) is the minimal distance between θ and a point of L.

These exponents were introduced originally by M. Laurent [START_REF] Laurent | On transfer inequalities in Diophantine approximation[END_REF]. They interpolate between the classical exponents ω(θ) = ω n-1 (θ) and λ(θ) = ω 0 (θ) (resp. ω(θ) = ωn-1 (θ) and λ(θ) = ω0 (θ)) that were introduced by A. Khinchin [START_REF] Ya | Über eine klasse linearer diophantischer approximationen[END_REF][START_REF] Ya | Zur metrischen theorie der diophantischen approximationen[END_REF], V. Jarník [START_REF] Jarník | Zum khintchineschen "Übertragungssatz[END_REF] and Y. Bugeaud and M. Laurent [START_REF] Bugeaud | On exponents of homogeneous and inhomogeneous diophantine approximation[END_REF][START_REF] Bugeaud | Exponents of diophantine approximation[END_REF].

We have the relations

ω 0 (θ) ≤ ω 1 (θ) ≤ • • • ≤ ω n-1 (θ), 1 ω0 (θ) ≤ ω1 (θ) ≤ • • • ≤ ωn-1 (θ),
and Minkowski's First Convex Body Theorem [START_REF] Minkowski | Geometrie der Zahlen[END_REF] and Mahler's compound convex bodies theory provide the lower bounds

ω d (θ) ≥ ωd (θ) ≥ d + 1 n -d , for 0 ≤ d ≤ n -1.
These exponents happen to be related, as was first noticed by Khinchin with his transference theorem [START_REF] Ya | Zur metrischen theorie der diophantischen approximationen[END_REF]. The study of these transferences has two aspects. First, establishing transference inequalities valid for every suitable point θ. Then, there is the reverse problem, that consists in constructing points θ to show that these inequalities are sharp. For this, one can prove that there exists points θ whose exponents satisfy the equality in the transference inequalities. In this case, we say that the inequalities are best possible. A stronger result is to prove that given k exponents e 1 , . . . , e k , the transference inequalities between these k exponents define a subset of R k that is exactly the set of all k-uples (e 1 (θ), . . . , e k (θ)) as θ runs through all points θ = (θ 1 , . . . , θ n ) ∈ R n such that 1, θ 1 , . . . , θ n are Q-linearly independent. The latter set is called the spectrum of the exponents (e 1 , . . . , e k ).

When the dimension is n = 1, we have the equality ω0 (θ) = ω(θ) = λ(θ) = 1. In [START_REF] Jarník | Zum khintchineschen "Übertragungssatz[END_REF], V. Jarník showed that in dimension n = 2, we have the following algebraic relation between ω1 (θ) and ω0 (θ):

ω0 (θ) + 1 ω1 (θ) = 1. ( * )
Furthermore, V. Jarník noted that, in higher dimension n ≥ 3, no algebraic relation holds anymore. He proved [6, Satz 3] that for n ≥ 2, there exist two n-tuples of real numbers θ = (θ 1 , . . . θ n ) and ν = (ν 1 , . . . , ν n ) such that ωn-1 (θ) = ωn-1 (ν) = +∞ , ω0 (θ) = 1 and ω0 (ν) = 1 n -1 .

V. Jarník also proved the following transference theorem:

Theorem 1 [START_REF] Jarník | Zum khintchineschen "Übertragungssatz[END_REF]. Let n ≥ 2. For any n-tuples of real number θ = (θ 1 , . . . θ n ) such that 1, θ 1 , . . . , θ n are Q-linearly independent, we have

ωn-1 (θ) (n -1)ω n-1 (θ) + n ≤ ω0 (θ) ≤ ωn-1 (θ) -n + 1 n .
If ωn-1 (θ) = n, the interval reduces to the single point ω0 (θ) = 1 n .

2

Remark. O. German [START_REF] German | On Diophantine exponents and Khintchine's transference principle[END_REF] and A. Khinchin [START_REF] Ya | On some applications of the method of the additional variable[END_REF] claim that V. Jarník [START_REF] Jarník | Zum khintchineschen "Übertragungssatz[END_REF] proved the existence of n-tuples θ = (θ 1 , . . . , θ n ) with ωn-1 (θ) = +∞ and ω0 (θ) anywhere in the interval 1

n -1 , 1 .
It appears to the author that this is not written explicitly in [START_REF] Jarník | Zum khintchineschen "Übertragungssatz[END_REF].

Recently, O. German [START_REF] German | On Diophantine exponents and Khintchine's transference principle[END_REF] improved Theorem 1:

Theorem 2 [START_REF] German | On Diophantine exponents and Khintchine's transference principle[END_REF]. With the notation of Theorem 1, we have

ωn-1 (θ) -1 (n -1)ω n-1 (θ) ≤ ω0 (θ) ≤ ωn-1 (θ) -(n -1) ωn-1 (θ) . ( * * )
Note that the interval reduces to a single point if n = 2, and that in this case we recover Jarník's relation ( * ).

The first goal of this paper is to prove that German's inequalities describe the spectrum of the two exponents (ω 0 , ωn-1 ) .

Theorem 3. Let n ≥ 2 be an integer, let ω ∈ [n, +∞] and let λ ∈ ω -1 (n -1)ω , ω -n + 1 ω ,
where we understand that the interval for λ is 1 n -1 , 1 when ω = +∞. Then there exist uncountably many n-tuples of real numbers θ = (θ 1 , . . . , θ n ), with 1, θ 1 , . . . , θ n Q-linearly independent, such that ωn-1 (θ) = ω and ω0 (θ) = λ.

In [START_REF] Schmidt | The generalization of jarnik's identity[END_REF], W. Schmidt and L. Summerer obtained independently a similar result, proving that the inequalities ( * * ) of German are best possible.

One can wonder if in higher dimension (n ≥ 3), there exists a Jarník's-type relation between the n uniform exponents ω0 , . . . , ωn-1 . The next theorem states that no such algebraic relation holds.

Theorem 4. For every integer n ≥ 3, the n uniform exponents ω0 , . . . , ωn-1 are algebraically independent.

Thus, the spectrum of the n uniform exponents ω0 , . . . , ωn-1 is a subset of R n with nonempty interior.

We also know the spectrum of other families of exponents. M. Laurent [START_REF] Laurent | Exponents of diophantine approximmation in dimension two[END_REF] described the spectrum of the four exponents ω 0 , ω0 , ω n-1 , ωn-1 in dimension n = 2. In his PhD thesis, the author gives an alternative proof of this result. However, for n ≥ 3 this spectrum is still unknown.

In this paper, we use the notation introduced by D. Roy in [START_REF] Roy | Spectrum of the exponents of best rational approximation[END_REF][START_REF] Roy | On Schmidt and Summerer parametric geometry of numbers[END_REF] which is essentially dual to the one of W. M. Schmidt and L. Summerer [START_REF] Schmidt | Parametric geometry of numbers and applications[END_REF][START_REF] Schmidt | Diophantine approximation and parametric geometry of numbers[END_REF]. We refer the reader to these papers for further details. Here x • y = x 1 y 1 + • • • + x n y n is the usual scalar product of vectors x and y, and

x 2 = √ x • x is the usual Euclidean norm.
Let u = (u 0 , . . . , u n ) be a vector in R n+1 , with Euclidean norm u 2 = 1. For a real parameter Q ≥ 1 we consider the convex body

C u (Q) = x ∈ R n+1 | x 2 ≤ 1, |x • u| ≤ Q -1 . For 1 ≤ d ≤ n + 1 we denote by λ d (C u (Q)) the d-th minimum of C u (Q) relatively to the lattice Z n+1 . For q ≥ 0 and 1 ≤ d ≤ n + 1 we set L u,d (q) = log λ d (C u (e q )) .
Finally, we define the following map associated with u:

L u : [0, ∞) → R n+1 q → (L u,1 (q), . . . , L u,n+1 (q)).
The lattice Z n+1 is invariant under permutation of coordinates. Hence, L u remains the same if we permute the coordinates in u. Since u 2 = 1 we can thus assume that u 0 = 0.

The following proposition links the exponents of Diophantine approximation associated with θ = ( u 1 u 0 , . . . , u n u 0 ) to the behavior of the map L u , assuming u 0 = 0. It was first stated by W.M. Schmidt and L. Summerer in [START_REF] Schmidt | Parametric geometry of numbers and applications[END_REF] (Theorem 1.4). It also appears as Relations (1.8) and (1.9) in [START_REF] Schmidt | Diophantine approximation and parametric geometry of numbers[END_REF]. In the notation of D.Roy [START_REF] Roy | Spectrum of the exponents of best rational approximation[END_REF] (Proposition 3.1), it reads as follows.

Proposition 1 [START_REF] Schmidt | Parametric geometry of numbers and applications[END_REF]. Let u = (u 0 , . . . , u n ) ∈ R n+1 , with Euclidean norm u 2 = 1 and u 0 = 0. Set θ = ( u 1 u 0 , . . . , u n u 0 ). For 1 ≤ k ≤ n, we have the following relations:

lim inf q→+∞ L u,1 (q) + • • • + L u,k (q) q = 1 1 + ω n-k (θ) , lim sup q→+∞ L u,1 (q) + • • • + L u,k (q) q = 1 1 + ωn-k (θ)
.

Thus, if we know an explicit map P = (P 1 , . . . , P n+1 ) : [0, ∞) → R n+1 , such that L u -P is bounded, then we can compute the 2n exponents ω0 (θ), . . . , ωn-1 (θ), ω 0 (θ), . . . , ω n-1 (θ) for the above point θ upon replacing L u,i by P i in the above formulas for 1 ≤ i ≤ n. For this purpose, we consider the following family of maps, introduced by D. Roy in [START_REF] Roy | Spectrum of the exponents of best rational approximation[END_REF].

Definition (Roy, 2014). Let I be a subinterval of [0, ∞) with non-empty interior. A generalized (n + 1)-system on I is a continuous piecewise linear map P = (P 1 , . . . , P n+1 ) : I → R n+1 with the following three properties.

(S1) For each q ∈ I, we have 0

≤ P 1 (q) ≤ • • • ≤ P n+1 (q) and P 1 (q) + • • • + P n+1 (q) = q. (S2) If H is a
non empty open subinterval of I on which P is differentiable, then there are integers r, r with 1 ≤ r ≤ r ≤ n + 1 such that P r , P r+1 , . . . , P r coincide on the whole interval H and have slope 1/(r -r + 1) while any other component P k of P is constant on H .

(S3) If q is an interior point of I at which P is not differentiable, if r, r, s, s are the integers for which

P ′ k (q -) = 1 r -r + 1 (r ≤ k ≤ r) and P ′ k (q + ) = 1 s -s + 1 (s ≤ k ≤ s) ,
and if r < s, then we have P r (q

) = P r+1 (q) = • • • = P s(q).
Here P ′ k (q -) (resp. P ′ k (q + )) denotes the left (resp. right) derivative of P k at q. The next result combines Theorem 4.2 and Corollary 4.7 of [START_REF] Roy | Spectrum of the exponents of best rational approximation[END_REF].

Theorem 5 ( Roy, 2014). For each non-zero point u ∈ R n+1 , there exists q 0 ≥ 0 and a generalized (n + 1)-system P on [q 0 , ∞) such that L u -P is bounded on [q 0 , ∞). Conversely, for each generalized (n + 1)-system P on an interval [q 0 , ∞) with q 0 ≥ 0, there exists a non-zero point u ∈ R n+1 such that L u -P is bounded on [q 0 , ∞).

In view of the remark following Proposition 1, this result reduces the determination of the joint spectrum of Diophantine approximation exponents to a combinatorial study of generalized (n + 1)-systems.

Although the definition of a generalized (n + 1)-system P = (P 1 , . . . , P n+1 ) may look complicated, it is easy to understand in terms of the combined graph of P , that is the union of the graphs of P 1 , . . . , P n+1 over the interval of definition I of P . We explain this below.

A division point of P is an endpoint of I contained in I or an interior point of I at which P is not differentiable. Such points form a discrete subset of I. Between two consecutive division points q * < q of I, the graph of each component of P is a line segment. All these line segments have slope 0 except for one line segment of positive slope 1/t where t is the number of components of P whose graph over [q * , q] is that line segment. In view of the condition P 1 ≤ P 2 ≤ • • • ≤ P n+1 , there must be consecutive components P r , . . . , P r of P with r -r + 1 = t. If q is also an interior point of I and if P s , . . . , P s are the components of P whose graph has positive slope 1 s -s + 1 to the right of q, then there are two cases.

1) If r < s, we say that q is an ordinary division point. In this case, we have P r (q) = • • • = P s(q) according to (S3). This implies that r ≤ s and r ≤ s. Among P r , . . . , P s, the components P j with s ≤ j ≤ r (if any) change slope from 1 r -r + 1 to 1 s -s + 1

. Those with j ≤ min(r, s -1) change slope from 1 r -r + 1 to 0. The remaining components P j with r + 1 ≤ j ≤ s -1 (if any) have constant slope 0 in a neighborhood of q. The reader is invited to draw a picture for himself or to look at those in [13, §4].

2) Otherwise, we have r > s because it cannot happen that r = s (or P is differentiable at q). Then, we say that q is a switch point. In this case, we have P r (q) = • • • = P r (q) > P s (q) = • • • = P s(q) which mean that the end point of the line segment of slope 1 r -r + 1 at the left of q lies above the initial point of the line segment of slope 1 s -s + 1 at the right of q.

It can be shown that the combined graph of a generalized (n + 1)-system P uniquely determines the map P provided that we know the value of P at one point of its interval of definition. An example of this is shown in [13, §4]. We will see two other examples in the sections 3 and 5.

In [START_REF] Schmidt | Parametric geometry of numbers and applications[END_REF][START_REF] Schmidt | Diophantine approximation and parametric geometry of numbers[END_REF] W. M. Schmidt and L. Summerer introduce the following exponents for an integer 1 ≤ d ≤ n + 1:

ϕ d = lim inf q→∞ L u,d (q) q , φd = lim sup q→∞ L u,d (q) q .
For these exponents, we have the following analogue of Theorem 4: Theorem 6. For every integer n ≥ 3, the exponents φ1 , . . . , φn are algebraically independent.

Proof of Theorem 3

In this section, we construct a family of generalized (n+1)-systems. Then, via Theorem 5, we get a family of n-tuples having the requested properties stated in Theorem 3. We first treat the case where ωn-1 is finite and n ≥ 3. We will explain later how to adapt the construction if n = 2 or ωn-1 is infinite.

First, note that a generalized (n + 1)-system with all components equal to q n + 1 provides via Theorem 5 a point θ with ωn-1 (θ) = n and ω0 (θ) = 1 n . Thus, we can exclude this case in the next construction.

Let q 0 be a positive real number, fix a real number ω > n ≥ 2 and set a parameter a with 1 n -1 ≤ a ≤ 1. We define the sequence (q 6m ) m≥0 by:

q 6m = (1 + a(ω -n))q 6(m-1) , for m ≥ 1.
Since ω > n, q 6m goes to infinity.

We construct a generalized (n + 1)-system P whose graph is invariant under the dilation of factor (1 + a(ω -n)) > 1 on the interval [q 0 , +∞). Thus, we only need to define P on a generic interval [q 6m , q 6(m+1) ]. Figure 1 shows the pattern of the combined graph of P .

For every integer m ≥ 0, we define P at q 6m as follows:

P 1 (q 6m ) = P 2 (q 6m ) = q 6m ω + 1 , P 3 (q 6m ) = • • • = P n (q 6m ) = 1 + ( 1 -a n -2 )(ω -n) ω + 1 q 6m , P n+1 (q 6m ) = 1 + a(ω -n) ω + 1 q 6m . P 2 P 3 = • • • = P n P n+1 P 1 P n+1 s l o p e 1 P 2 P 1 s l o p e 1 s l o p e 1 n -2 P 3 = • • • = P n q 6m q 6m+1
q 6m+2 q 6m+3 q 6m+4 q 6m+5 q 6(m+1)

Figure 1: Combined graph of P on a generic interval [q 6m , q 6(m+1) ]

Here the parameter a says how large P n+1 is at each point q 6m . The condition a ≥ 1 n -1 imposes the condition P n+1 (q 6m ) ≥ P n (q 6m ), and the condition a ≤ 1 imposes that P 3 (q 6m ) ≥ P 2 (q 6m ). We have the dilation condition P (q 6(m+1)

) = P ((1 + a(ω -n))q 6m ) = (1 + a(ω -n))P (q 6m
) by the definition of the sequence (q 6m ) m≥0 .

For k = 0, . . . , 5 the graph has only one line segment of positive slope on the interval [q 6m+k , q 6m+k+1 ]. The graph is clearly the combined graph of a generalized (n + 1)-system with seven division points q 6m , . . . , q 6m+6 . The points q 6m+3 and q 6m+5 are switch points while the others are ordinary division points. Furthermore it is uniquely defined since we know the value of P at the point q 6m , where as requested

P 1 (q 6m ) + • • • + P n+1 (q 6m ) = q 6m .

Easy computation gives

q 6m = (1 + a(ω -n))q 6(m-1) , q 6m+1 = (n -2)(ω + 1) + (1 -a)(ω -n) (n -2)(ω + 1) q 6m , q 6m+2 = (n + 1) + (1 + a)(ω -n) ω + 1 q 6m , q 6m+3 = ω + (1 + a(ω -n)) 2 ω + 1 q 6m , q 6m+4 = 1 + (1 + a(ω -n))(n + a(ω -n) ω + 1 q 6m , q 6m+5 = 1 + 2a(ω -n) + ω(1 + a(ω -n)) ω + 1 q 6m .
We now compute its associated exponents with Proposition 1. One can notice that the local extrema of the functions q → q -1 P k (q), 1 ≤ k ≤ n + 1 are located at division points where P k changes slope.

Since P is invariant under dilation of factor C = (1 + a(ω -n)) we have for every m ≥ 0, every 1 ≤ k ≤ n + 1, and every q in [q 6m , q 6m+6 ) the relation

q -1 P k (q) = q -1 C m P k qC -m ,
where C -m q lies in the fundamental interval [q 0 , q 6 ]. Thus, lim sup q→+∞ P 1 (q) q = max q 0 ≤q≤q 6

P 1 (q) q = P 1 (q 0 ) q 0 = 1 ω + 1 , lim inf q→+∞ P n+1 (q) q = min q 0 ≤q≤q 6 P n+1 (q) q = P n+1 (q 2 ) q 2 = 1 + a(ω -n) n + 1 + (1 + a)(ω -n) ,
because the component P n+1 changes slope from zero to some positive value only at q 6m+2 .

Then, according to Proposition 1, Theorem 5 provides an n-tuple θ = (θ 1 , . . . , θ n ) such that 1 ωn-1 (θ) + 1 = lim sup

q→+∞ P 1 (q) q = 1 ω + 1 , ω0 (θ) ω0 (θ) + 1 = lim inf q→+∞ P n+1 (q) q = 1 + a(ω -n) n + 1 + (1 + a)(ω -n) .
Thus, this θ satisfies ωn-1 (θ) = ω and ω0 (θ

) = 1 + a(ω -n) ω .
When a runs through the interval [ 1

n -1 , 1], then ω0 (θ) runs through the interval ω -1 (n -1)ω , ω -(n -1) ω .
If n = 2, we remove the line P 3 = • • • = P n and the interval [q 6m+3 , q 6m+5 ] from the generic graph on the interval [q 6m , q 6(m+1) ], the parameter a is then forced to be equal to 1. Thus, we construct θ with ω1 (θ) = ω and ω0 (θ) = 1 -1 ω, which agrees with Jarník's relation ( * ).

If ω is infinite, we replace ω by m + n + 1 in our construction. For a given real number q 0 we consider the sequence (q 6m ) m≥1 defined by q 6m = (m + 1)q 6(m-1) .

Figure 1 still represents the combined graph from P on a generic interval [q 6m , q 6m+6 ], with the following settings at q 6m :

P 1 (q 6m ) = P 2 (q 6m ) = q 6m m + n + 2 , P 3 (q 6m ) = • • • = P n (q 6m ) = 1 + ( 1 -a n -2 )(m + 1) m + n + 2 q 6m , P n+1 (q 6m ) = 1 + a(m + 1) m + n + 2 q 6m .
Note that the combined graph is not invariant under dilation anymore. We have lim sup q→+∞ P 1 (q) q = lim sup m→+∞ max q 6m ≤q≤q 6(m+1)

P 1 (q) q = lim sup m→+∞ P 1 (q 6m ) q 6m = lim sup m→+∞ 1 m + n + 2 = 0, lim inf q→+∞ P n+1 (q) q = lim inf m→+∞ min q 6m ≤q≤q 6(m+1) P n+1 (q) q = lim inf m→+∞ P n+1 (q 6m+2 ) q 6m+2 = lim inf m→+∞ 1 + a(m + 1) n + 1 + (1 + a)(m + 1) = a a + 1 .
Again, Theorem 5 provides us with an n-tuple θ = (θ 1 , . . . , θ n ) such that ωn-1 (θ) = +∞ and ω0 (θ) = a, where a runs through the interval [ 1

n -1 , 1].
Note that if 1, θ 1 , . . . , θ n are Q-linearly dependent, then there exists an integer point x ∈ Z n such that |x • u| = 0. This implies that L u,1 (q) is bounded above by log( x 2 ). In our construction by dilatation P 1 is not bounded, hence the independence by contradiction.

To complete the proof of Theorem 3, we have to check that we can construct uncountably many n-tuples with given exponents. Let ω and λ as in Theorem 3, and a the parameter such that Theorem 5 provides an n-tuple θ whose exponents satisfy ωn-1 (θ) = ω, and ω0 (θ

) = λ = 1 + a(ω -n) ω .
Fix q 0 a real number to start the construction from P as above with parameter a. For every ρ 1 and ρ 2 such that q 0 ≤ ρ 1 < ρ 2 ≤ q 5 , we denote by P ρ 1 and P ρ 2 the (n + 1)-generalized system with parameter a starting in ρ 1 and ρ 2 . We have P ρ 1 (q 6 ) = P ρ 2 (q 6 ) and

P ρ 1 (q 6m ) -P ρ 2 (q 6m ) ∞ = q 6m q 6 P ρ 1 (q 6 ) -P ρ 2 (q 6 ) ∞ → n→∞ ∞, where (x 1 , . . . , x n ) ∞ = max 1≤k≤n |x k |.
Thus, their difference is unbounded, and they cannot correspond to the same θ via Theorem 5. This ends the proof of Theorem 3.

An alternative proof of Theorem 2

In this section, we give an alternative proof of Theorem 2 using arguments from Parametric Geometry of Numbers.

One can notice that the extremal values of the components of P are reached at the division points. The condition (S3) translates into the fact that for every division point q, the right endpoint of the segment with non-zero slope ending at q lies above the left endpoint of the one starting at q. A first consequence is that when P 1 is non constant, it increases until reaching P 2 (q). A second consequence is the following proposition. Proposition 2. For every 1 ≤ k < m ≤ n + 1, if p 0 is a division point such that P k has slope changing from 0 to 1 at p 0 , then we have for every p > p 0

P m (p) ≤ max(P m (p 0 ), P k (p 0 ) + p -p 0 ).
In particular, P m is constant on the interval [p 0 , p 0 + P m (p 0 ) -P k (p 0 )].

P k+1 P m P k p 0 + P m (p 0 ) -P k (p 0 ) p 0
Upper bound: Let P be a generalized (n + 1)-system, and ω the real number such that lim sup q→+∞ P 1 (q) q = 1 ω + 1 .

If this limit is zero, we set ω = +∞. We treat this case after.

Let ε > 0. There exist arbitrarily large division points p 0 where q -1 P 1 (q) has a local maximum and

1 -ε ω + 1 ≤ P 1 (p 0 ) p 0 ≤ 1 + ε ω + 1 .
Since q 0 is a local maximum, we have P 1 (q 0 ) = P 2 (q 0 ). Furthermore, P 1 (q) ≤ P 2 (q) ≤ • • • ≤ P n+1 (q) and P 1 (q) + • • • + P n+1 (q) = q provide

P n+1 (p 0 ) ≤ p 0 -nP 1 (p 0 ) ≤ ω + 1 -n -nε ω + 1 p 0 .
At the point p = p 0 + ω -n -nε ω + 1 p 0 , according to Proposition 2, we have the upper bound

P n+1 (p) ≤ max(P n+1 (p 0 ), P 1 (p 0 ) + p -p 0 ) ≤ 1 + ε + ω -n -nε ω + 1 p 0 .
Thus, for arbitrarily large real numbers p, we have

P n+1 (p) p ≤ ω + 1 -n -(n -1)ε 2ω -n + 1 -nε , thus λ λ + 1 = lim inf q→+∞ P n+1 (q) q ≤ ω + 1 -n 2ω -n + 1 , giving that λ ≤ ω -(n -1) ω .
If ω is infinite, then we have

0 ≤ P 1 (p 0 ) p 0 ≤ ε , P 1 (p 0 ) = P 2 (p 0 ).
Using Proposition 2 at the point p = 2p 0 -nε we get

P n+1 (p) ≤ p 0 -(n + 1)ε.
Thus, for arbitrarily large real numbers p, we have

P n+1 (p) p ≤ p 0 -(n + 1)ε 2p 0 -nε .
This implies that λ ≤ 1.

Thus, we have proved the upper bound in Theorem 2.

Lower bound:

Let P be a generalized (n + 1)-system, and ω, λ the real numbers such that lim sup

q→+∞ P 1 (q) q = 1 ω + 1 , lim inf q→+∞ P n+1 (q) q = λ λ + 1 ,
where we understand that ω = +∞ if the corresponding lim sup is zero. Again, we treat this case after.

Let ε 1 > 0. There exists a real number q 0 such that q ≥ q 0 implies

P 1 (q) q ≤ 1 + ε 1 ω + 1 .
Let ε 2 > 0. There exist arbitrarily large division points p ≥ q 0 where q -1 P 1 (q) has a local minimum and

P n+1 (p) p - λ λ + 1 ≤ ε 2 .
Let p 0 = max {q ≤ p | P 1 (q) = P 2 (q)}. At the point p 0 we have

P 1 (p 0 ) = P 2 (p 0 ) ≤ 1 + ε 1 ω + 1 p 0 , P n+1 (p 0 ) ≥ p 0 -2P 1 (p 0 ) n -1 , since p 0 = P 1 (p 0 ) + • • • + P n+1 (p 0 ) ≤ 2P 1 (p 0 ) + (n -1)P n+1 (p 0 ).
We first show that q → P 1 (q) is constant on the interval [p 0 , p]. If not, there exists a real number p 0 < p 1 < p where P 1 has slope 1. Since p is a local minimum from q -1 P 1 (q), we have P ′ 1 (p -) = 0. Thus, there exists a point in (p 1 , p) where P 1 changes slope from 1 to 0. At this point P 1 = P 2 , which contradicts the definition of p 0 . Thus,

P 1 (p 0 ) = P 1 (p).
We can write

p = n+1 k=1 P k (p) ≤ nP n+1 (p) + P 1 (p 0 ). Thus, P n+1 (p) p ≥ P n+1 (p) nP n+1 (p) + P 1 (p 0 ) ,
where the right hand side is an increasing function of P n+1 (p). Since

P n+1 (p) ≥ P n+1 (p 0 ) ≥ p 0 -2P 1 (p 0 ) n -1 ,
we have

P n+1 (p) p ≥ p 0 -2P 1 (p 0 ) np 0 -(n + 1)P 1 (p 0 ) ,
where the right hand side is a decreasing function of P 1 (p 0 ). Since

P 1 (p 0 ) ≤ 1 + ε 1 ω + 1 p 0 ,
we have

P n+1 (p) p ≥ ω -1 -2ε 1 nω -1 -(n + 1)ε 1 . Thus, λ λ + 1 ≥ ω -1 -2ε 1 nω -1 -(n + 1)ε 1 -ε 2 .
This gives the expected bound λ ≥ ω -1 (n -1)ω .

If ω is infinite, we consider a real number q 0 such that q ≥ q 0 implies 0

≤ P 1 (q) q ≤ ε 1 .
This provides the following estimates at p:

P n+1 (p) p ≥ 1 -2ε 1 n -(n + 1)ε 1 .
Thus, we get λ

λ + 1 ≥ 1 -2ε 1 n -(n + 1)ε 1 -ε 2 .
This gives the expected bound λ ≥ 1 n -1 .

Thus, we have proved the lower bound. This ends the proof of Theorem 2.

Proof of Theorems 4 and 6

In this section, we construct a family of generalized (n + 1)-systems depending on n parameters which via Theorem 5 provides us with a family of n-tuples whose uniform exponents are expressed as a function of these n parameters. Then, we show that these functions are algebraically independent.

Fix the dimension n ≥ 3. Choose n + 2 parameters A 1 , A 2 , . . . , A n+1 , C satisfying

0 < A 1 = A 2 < A 3 < A 4 < • • • < A n+1 , 1 = A 1 + A 2 + • • • + A n+1 , A k+1 A k < C < A k+2 A k for 2 ≤ k ≤ n -1, 1 < A n+1 A n < C. ( 0
)
We consider the generalized (n + 1)-system P on the interval [1, C] whose combined graph is given by Figure 2, where On each interval between two consecutive division points, there is only one line segment with nonzero slope. This line segment has slope 1 on the intervals [1, δ 2,1 ], [δ n+1,1 , C] and [δ k-1,2 , δ k,1 ] for 3 ≤ k ≤ n + 1, and has slope 1/2 on the interval [δ k,1 , δ k,2 ] , for 3 ≤ k ≤ n , where the two components P k and P k+1 coincide. We have 2n + 1 division points 1, C, δ k,1 and δ l,2 for 2 ≤ k ≤ n + 1 and 2 ≤ l ≤ n. They are all ordinary division points except δ n+1,1 which is a switch point. Note that the conditions (0) are consistent with the graph. The points which will be most relevant for the proofs are labeled with black dots.

P k (1) = A k and P k (C) = CA k for 1 ≤ k ≤ n + 1. • • • • • • • • • • • • • • • • • • A 1 = A 2 A 3 CA 2 A 4 A k P k A k+1 P k+1 A n+1 P n+1 P k-1 P k+1 P n P n+1 CA n+1 CA n P 1 P 2 P k-1 CA k-1 P k 1 C δ k-1,1 δ k-1,2 δ k,1 δ k,2 δ 2,1 δ 2,2 δ 3,1 δ n+1,1
We extend P to the interval [1, ∞) by self-similarity, that is P (q) = C m P (qC -m ) for every positive integer m. In view of the value of P and its derivative at 1 and C, one sees that this extension provides a generalized (n + 1)-system on [1, ∞). Since Ŵ0 , . . . , Ŵn-1 come from a generalized (n + 1)-system P , Theorem 5 provides a point θ in R n such that ωk (θ) = Ŵk for every 0 ≤ k ≤ n -1. Thus, to prove Theorem 4, it is sufficient to show that the rational fractions Ŵ0 , . . . , Ŵn-1 ∈ Q(C, A 2 , A 3 , . . . , A n ) are algebraically independent.

Suppose on the contrary that there exists an irreducible polynomial R ∈ Q(X 1 , . . . , X n ) such that R Ŵ0 , Ŵ1 , . . . , Ŵn-1 = 0.

Specializing C in 0, we obtain

R 1 -A 2 -A 2 -• • • -A n A 2 , 1 -A 2 -A 2 -• • • -A n A 2 , . . . , 1 -A 2 -A 2 -A 3 A 2 , 1 -A 2 A 2 = 0.
Here, the first two rational fractions are the same, and the last n -1 rational fractions generate the field Q(A 2 , A 3 , . . . , A n ). Therefore the latter are algebraically independent, and R = α(X 2 -X 1 ) for a nonzero constant α ∈ Q. This is impossible since Ŵ0 = Ŵ1 .

Proof of Theorem 6

We consider the same generalized (n + 1)-system as above. Notice that for 1 ≤ k ≤ n we have P k ≤ P n+1 and therefore 0 ≤ P k (q) q ≤ 1/2.

Since all nonzero slopes of the combined graph are at least 1/2, the maxima of the functions q → q -1 P k (q) are reached at points where P k changes slope from 1 or 1/2 to 0. It happens that for each component there is only one such point on the interval [1, C[. The definition of the exponents φk leads to define quantities F k by

F k := lim sup q→∞ P k (q) q = max [1,C] P k (q) q = P k (p) p where p = 1 if k = 1, δ k,2 if 2 ≤ k ≤ n.
We express the quantities F 1 , . . . F n as rational fractions in Q(C, A 2 , . . . , A n ), using the relations

A 1 = A 2 and A n+1 = 1 -A 1 -A 2 -• • • -A n : F 1 = A 1 , F k = CA k A 1 + C(A 2 + • • • + A k ) + CA k + 1 -(2A 2 + A 3 + • • • + A k+1 ) ,
Since F 1 , . . . , F n come from a generalized (n + 1)-system P , by Theorem 5 there exists a point θ in R n such that φk (θ) = F k for every 1 ≤ k ≤ n. To prove Theorem 6 it is sufficient

Figure 2 :

 2 Figure 2: Pattern of the combined graph of P on the fundamental interval [1, C]

Acknowledgements

The author would like to thank the referee and Damien Roy for careful reading and useful remarks to simplify and shorten the proofs.

Proposition 1 suggests to define quantities Ŵn-1 , . . . , Ŵ0 by 1 1 + Ŵn-k := lim sup

Since P is invariant under dilation of factor C, we can replace lim sup q→∞ by max [1,C] in the above formulae.

We observe that for 1 ≤ k ≤ n, the function

]. Thus the maximum on [1, C] of the function q → q -1 (P 1 (q) + • • • + P k (q)) is reached either at δ k,1 or at δ k,2 , when slope changes from 1 to 1/2 or from 1/2 to 0. Namely, the maximum is reached at δ k,1 if

and at δ k,2 if the lefthand side is ≤ 1/2. We deduce that for 1 ≤ k ≤ n,

For 2 ≤ k ≤ n + 1, we have the following values at δ k,1 and δ k,2 :

It is easy to check that the parameters

satisfy the conditions (0). For this choice of parameters, the lefthand side of inequality (2) is > 1/2 for 1 ≤ k ≤ n -1 and < 1/2 for k = n. This property remains true for (C, A 2 , . . . , A n ) in an open neighborhood of (3, 2 -n , . . . , 2 -2 ) provided that we set

In this neighborhood, the quantities Ŵ0 , . . . , Ŵn-1 are given by the following rational fractions in Q(C, A 2 , A 3 , . . . , A n ) :

to show that the rational fractions F 1 , . . . , F n ∈ Q(C, A 2 , A 3 , . . . , A n ) are algebraically independent.

Suppose that there exists an irreducible polynomial R ∈ Q(X 1 , . . . , X n ) such that

Specializing C in infinity, we obtain

where all coordinates except 1/2 are algebraically independent. Thus, R is a constant multiple of 2X 2 -1, which contradicts F 2 = 1/2.

We are not able to prove Theorem 6 for the n + 1 exponents φ1 , . . . , φn+1 with this construction. However with some extra work, we can show that the theorem holds for any n exponents among them.


J.-Ph Brazier

M Huet

O Léon

M Itasse

MODAL ANALYSIS OF JET NEAR-FIELD PRESSURE FLUCTUATION

Keywords: jet noise, LES, instability, PSE, POD

Purpose

Unstable Kelvin-Helmholtz waves are suspected to be responsible for a large part of low-frequency noise radiation in high-speed jets. The purpose of this paper is to check the coherence of several numerical and experimental data concerning this phenomenon, in the particular case of a cold subsonic jet.

Design/methodology/approach

In the present work, a cross investigation of the near pressure field is performed on three different data sets: LES computations, PSE semi-modal computations and microphone measurements, in order to determine the local amplitudes of unstable waves.

Findings

The large coherent structures are found in both LES and experimental results and they are also in good agreement with direct semi-modal computations carried out with the PSE approach.

Originality/value

This work confirms that the unstable wave packets can be extracted from both LES and experimental results, provided that an appropriate modal decomposition is performed.

static temperature in the centre of the nozzle exit section U j axial velocity in the centre of the nozzle exit section U x axial velocity for the jet base flow x axial coordinate  axial wavenumber ω angular frequency

Introduction

Understanding the physical mechanisms of jet noise has been the subject of many works and articles for several decades and in spite of important progress in both experimental and numerical domains, this topic is still relevant. In particular, beside small-scale chaotic turbulence, large-scale coherent structures have been observed in jets mixing layers [START_REF] Crow | Orderly structure in jet turbulence[END_REF]. These large coherent structures are generated by Kelvin-Helmholtz instability waves which are amplified at the nozzle exit, due to the strong shear in the mixing layer, and then damped further downstream [START_REF] Tam | Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets[END_REF]. They can be represented as wave packets of limited axial extent. Their role in the generation of directional low-frequency noise is more and more suspected, according to theoretical modelling as well as experimental investigation [START_REF] Suzuki | Instability waves in a subsonic round jet detected using a near-field phased microphone array[END_REF]. A good review can be found in the recent article by [START_REF] Jordan | Wave packets and turbulent jet noise[END_REF] If the acoustic radiation of supersonic jets can explained by a mechanism based on Mach waves [START_REF] Troutt | Experiments on the flow and acoustic properties of a moderate-Reynolds-number supersonic jet[END_REF], the extension to subsonic jets is not obvious. However, it has been shown in recent works that the sound directivity of a subsonic jet could be favourably compared with the acoustic radiation of a model wave packet [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF]. In hot subsonic jets, due to the temperature ratio between the jet and the ambient medium, some coherent structures can be convected at supersonic speeds with regard to the outer sound celerity. In this case, the sound radiation mechanism is similar to that of supersonic jets. But this cannot be invoked for cold subsonic jets and the first purpose of the present work was to investigate particularly the pressure near-field of such a cold subsonic jet, in order to feed a future acoustic investigation.

Derived from linearised Euler or Navier-Stokes equations, the Parabolised Stability Equations (PSE) can be seen as an extension of the local linear stability approach, since they can take into account the slow axial variation of the base flow, and they constitute an efficient way to compute the axial evolution of an unstable wave packet. They were first derived to compute boundary-layer instabilities [START_REF] Herbert | Parabolized Stability Equations[END_REF] and then applied to jets [START_REF] Yen | Application of parabolized stability equations to the prediction of jet instabilities[END_REF]Messersmith, 1998, Balakumar, 1998). A close agreement between PSE results and near-field pressure measurements made with circular antennas was reported [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF]. However, the far-field acoustic radiation cannot be directly provided by the PSE solution and an extrapolation with a Kirchhoff integral formula is required [START_REF] Balakumar | Prediction of supersonic jet noise[END_REF]. Dual-stream jets were also investigated [START_REF] Léon | Investigation of the near and far pressure fields of dual-stream jets using an Euler-based PSE model[END_REF]. The PSE can also be extended to deal with weakly nonlinear interactions between unstable modes (Itasse et al., 2015 and references included).

In the framework of the ORINOCO project, carried out in cooperation between European and Russian laboratories, concurring computations and experiments were performed on a cold subsonic jet. Using three different numerical and experimental data sets collected at ONERA during this project, it is possible to explore the near pressure field of this jet and to verify that the signature of the large coherent structures can be extracted from measurements and correctly predicted by computations. This is the second purpose of the present paper.

Experiment

Experimental setup 2.1.

The cold subsonic jet investigated during the ORINOCO project was produced by a circular nozzle of diameter D = 50 mm. The geometry of this nozzle had been formerly designed during the European project JEAN and the nozzle lip has been thickened to 15 mm to integrate plasma actuators. In the present case, only the unforced jet is considered. In the exit section, the flow on the nozzle axis is characterised by a Mach number M = 0.85, an axial velocity U j = 292 m/s and a static temperature T j = 296 K. Outside of the jet, the parameters of the quiet air are a pressure of 1 atm and a temperature of 295 K, giving a sound speed of 344 m/s.

In the following, the frequency will often be replaced by the dimensionless Strouhal number, defined as

St = f D U j ( 1 )
where D is the nozzle diameter. The instabilities of a circular jet are typically encountered for Strouhal values between 0.1 and 0.9, with a maximum near 0.4. In the case of the present jet, it corresponds to frequencies roughly lying between 600 Hz and 5000 Hz. Several measurement techniques were used to investigate both the mean flow field and the fluctuating part of this jet. First, the mean velocity field was measured with a two-component PIV system in the axial sections. The PIV system was composed of a MotionPro X5 camera of 2352 x 1728 px resolution mounted with a 50 mm lens, and a double-pulsed Nd:YAG Quantel laser of 150 mJ per pulse. The system was operated at a repetition rate of 4 Hz, with an inter-pulse period of 2 µs. Atomized DEHS oil, providing droplets with a mean diameter less than 1 µm, was employed to seed the flow. The seeding was introduced upstream of the jet exit, at about 30 D and before the flow rectifier located in the air-supplying pipe. This ensured a homogeneous seeding in the jet core. Furthermore, seeding was also introduced around the exit plane, at very low velocities, in order to seed the surrounding air dragged along by the jet. In order to accurately measure the entire flow field up to 10 D downstream, the PIV system was mounted on a traverse system and a set of eight spatial windows was acquired with overlaps. For each acquisition window, a total of 400 pairs of images was acquired, which was observed to be sufficient to provide well-converged first and second-order moment statistics. Post-processing was performed based on a classical cross-correlation technique using LaVision's Davis software, with interrogation windows of 16 px x 16 px, leading to a spatial resolution of 0.81 mm.

Then, near-field pressure fluctuations were recorded using a linear microphone array located close to the jet axis (Figure 1). This array had 14 microphones and formed an angle of 11 degrees with the jet axis. The first microphone was located at an abscissa x = 2 D and a radius r = 1.15 D. In this way, the array was located in the linear-hydrodynamic region [START_REF] Suzuki | Instability waves in a subsonic round jet detected using a near-field phased microphone array[END_REF], where the pressure fluctuations are dominated by the instability waves, whereas farther from the jet, the pressure fluctuations are driven by the acoustic field only. The sampling frequency was 48 kHz.

Navier-Stokes computations

Mean flow computation 3.1.

Before performing the PSE analysis, the jet mean flow field must be computed. This was done using ONERA code CEDRE, which solves the Navier-Stokes equations on an unstructured grid with a finite-volume formulation. The steady solution (Figure 1) was obtained by a RANS computation using a k-l turbulence model on a three-dimensional grid corresponding to a 15 degrees angular sector. The full grid was composed of 2.2 x 10 6 elements; the fluid volume was discretized with tetrahedra while prisms were used at the nozzle wall to capture the turbulent boundary-layer, together with a turbulent wall-law model. Special care was given to the resolution of the flow gradients in the shear layer with the use of very small elements. Spatial discretization was performed with a second-order MUSCL scheme and time integration was achieved with a first-order implicit scheme, allowing the use of a large time step to fasten the convergence of the mean flow. Finally, inlet turbulent parameters were tuned to best fit the experimental results and the values imposed at nozzle inlet were k = 5 m 2 .s -2 and l = 0.005 m. They correspond to a homogeneous isotropic turbulence with velocity fluctuations amplitudes being 3% of the mean inlet velocity and a length scale of 1/20 th of the inlet boundary diameter. As it was not possible to recover with the simulation both the potential core length and the correct downstream axial velocity decrease provided by PIV, a dilatation coefficient had to be applied to the axial coordinate in the computation. The turbulence level in the nozzle exit section has been tuned in such a way that after dilatation, the length of the potential core and the decrease of the velocity on the jet axis both agreed with experiment (Figure 2). A satisfying agreement was also observed between computation and PIV measurements on the radial profiles of axial velocity, which is mandatory for shear layer instability calculations. The unsteady jet simulation was also performed with CEDRE. A second-order scheme was used for the spatial discretization, together with a first-order implicit scheme for time integration. A Smagorinsky subgridscale model represented the filtered structures in the frame of Large-Eddy Simulation (LES).

Simulations have been performed with a time step ∆t = 3×10 -7 s which ensured a CFL number lower than 1 in almost all the numerical domain. After a transient period of 24 ms (80,000 iterations) required to let the jet flow develop, the simulations were run for 234,000 iterations to perform mean flow averaging and far-field noise radiation. It corresponds to a physical time of 70.20 ms or to a non-physical time of 420 convective time units D / Uj. Previous simulations of a similar case [START_REF] Huet | Simulation of flow control with microjets for subsonic jet noise reduction[END_REF] have demonstrated that this duration is sufficient to provide a statistical convergence of the flow fields and far-field pressure spectra.

The grid has been shown to be fine enough to accurately propagate acoustic waves at least up to a Strouhal number of 1.15 (corresponding to a frequency of 7000 Hz) with 30 points per wavelength. To ensure a convenient turbulence level in the nozzle exit section, the turbulence must be triggered in the nozzle internal boundary-layer. This was done through a geometrical disturbance of the nozzle inner wall, from x / D = -0.9 to x / D = -0.5, with an amplitude of ± 0.02 D (1 mm).

For data post-processing, the time results were stored on several cylindrical or conical surfaces. One of them was intended for Kirchhoff integral extrapolation of the acoustic far field, which will not be presented here. Its position is shown in Figure 1. Unfortunately, it does not coincide with the position of the linear antenna, but PSE computations will make possible some comparisons between both surfaces. A more detailed presentation of the unsteady simulations and associated results is available in [START_REF] Huet | On the use of Plasma Synthetic Jets for the control of jet flow and noise[END_REF].

PSE computations

The Parabolised Stability Equations (PSE) approach has been widely applied first to boundary-layer instability and then to jet instability [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF][START_REF] Léon | Investigation of the near and far pressure fields of dual-stream jets using an Euler-based PSE model[END_REF]. This method can be employed to describe wave propagation in flows with a strong transverse variation and a slow axial evolution, which is the case for circular jets. Starting from linearised perturbation equations, the time and azimuthal dependences are expressed through an exponential term whereas the axial evolution is split in two parts, according to the scheme: ϕ x, r, θ, t ϕ x, r . e imθ -iωt . e A(x)

(2)

with A x iα ξ dξ .

In the above expression, ω represents the angular frequency and m is the integer azimuthal wave number. The real part of the complex parameter α represents the local axial wave number whereas the imaginary part of α is the opposite of the local amplification rate. In this way, all the rapid phase fluctuations and the exponential growth should be included in A(x) and therefore the amplitude function ϕ should only exhibit slow variation in the axial direction. The introduction of α must be compensated by a closure relation called the normalisation condition. This relation is not unique but the most common one will be used in the present work:

ϕ * dr = 0

( 3 )

For jet instability, the viscosity does not play a key role and it can be neglected as a first step. The above decomposition (Eq. 2) is thus introduced in Linearised Euler Equations, with prescribed frequency and azimuthal wave number. The resulting set of equations can be shown to be quasi-parabolic in the axial direction and therefore, due to the slow evolution of the amplitude variables, it can be integrated at low computational cost by a marching procedure with a large step size. At each step, the normalisation condition (Eq. 3) is solved iteratively. The initial solution at x = 0 is provided by a local stability analysis. For linear PSE computations, the initial amplitude is arbitrary and it is fixed by imposing the maximum of pressure fluctuation equal to unity. The true amplitude level will be determined later by comparison with LES or experimental results.

Comparison of LES and PSE results

Processing of LES results

5.1.

The amplitude of pressure fluctuations predicted by LES and PSE approaches will be compared along the line representing the Kirchhoff's surface (Figure 1). First, the temporal LES results have been processed with a double Fourier transform on both time and azimuthal coordinates. The diagram of power spectral density for the pressure fluctuation given by LES on the Kirchhoff's conical surface is plotted in Figure 5 for azimuthal wave numbers m = 0 (left) and m = 1 (right). It can be observed that except for very low frequencies, the power density increases until x / D close to 3 and then decreases. The maximum is obtained for Strouhal numbers between 0.2 and 0.6, which corresponds roughly to the range of the most 4). Moreover, the density is slightly higher for m = 0 than for m = 1. The high levels on low frequencies past x / D = 5 may be due to insufficient statistical convergence of the LES downstream, caused by the limited time length of the simulation.

Figure 6 shows the repartition of the power between azimuthal modes for a fixed frequency, corresponding to St = 0.4, still along the conical surface. The comparison between positive and negative helical modes, which should have the same level, gives an appraisal of the precision of the results. Up to x / D = 4, the axisymmetric mode dominates and modes with |m| > 2 are far lower.

PSE modes fitting on LES results

5.2.

From now on, the amplitudes are plotted in dB logarithmic scale and therefore a multiplicative coefficient on the initial amplitude corresponds to a vertical translation of the curves. The pressure fluctuations along the conical surface are plotted for Strouhal numbers from 0.3 to 0.6 and for azimuthal wave numbers m = 0 and m = 1 in Figures 7 and8. In these cases, the initial exponential growth and the maximum of fluctuation are very well reproduced by the PSE solutions, but downstream they decrease strongly whereas the LES levels exhibit constant or slowly decreasing values. A possible explanation to this difference will be proposed in the next paragraph when a comparison between PSE and experimental levels will be proposed.

The same kind of results has been obtained on the other cylindrical surfaces R = 0.3 D, 0.5 D and 1.2 D where LES data were also stored, but they will not be all presented here. An example of amplitude adjustment for the PSE on the cylindrical surface R = 0.5 D is shown in Figures 14 and15 in the Appendix. The important fact is that the initial amplitude of a given PSE mode remains of the same order of magnitude when LES and PSE data are fitted on different surfaces. 1). First, the Power Spectral Density (PSD) has been computed in the frequency domain for each microphone. A circular antenna would have been necessary to provide information on the azimuthal structure of the pressure fluctuations, but such a device was unfortunately not available. However, in the LES data set, the axisymmetric mode m = 0 has been seen to dominate the helical modes in the first part of the jet. Therefore, as a first approximation, the PSE axisymmetric mode alone will be compared to the measurements.

Moreover, since all the microphones have been recorded simultaneously, it has been possible to perform a Proper Orthogonal Decomposition (POD) on the antenna signal. This statistical analysis allows to isolate the most coherent structures in the axial direction. In this way, the adjustment of PSE amplitudes on experimental values is shown in Figures 9 and 10 for Strouhal numbers from 0.3 to 0.6. It appears that the first microphone is located too much downstream to provide information on the exponential growth phase of the modes, close to the nozzle. However, the first POD mode compares favourably with the decrease stage of the PSE mode, whereas the global PSD of the antenna stagnates or decreases slowly, in the same way that was observed for LES data in the previous paragraph. Applying the same POD algorithm to the previous LES results would perhaps allow to isolate axially coherent structures with a decrease slope closer to that of the PSE modes. This has been carried out for the LES data on the conical surface. The first axial POD modes computed from LES results are plotted in Figures 11 and12, together with PSE results, for Strouhal numbers from 0.3 to 0.6 and for the azimuthal wave numbers m = 0 and m = 1. Obviously, the POD method applied to LES results does not provide smooth results, particularly downstream of x/D = 6. This was already observed on azimuthal decomposition (Figures 6 to 8) and it is probably due to an insufficient time convergence of the computation in this region. However, the strong decrease of the first POD modes is comparable to that of PSE curves and it is probably representative of a physical phenomenon since the same tendency is observed on experimental results (Figures 9 and10). This qualitative result should be confirmed with more accurate LES simulations.

Comparison of experimental and measured amplitudes

The comparison of the absolute amplitudes of PSE modes fitted on LES and experimental data is summarized in Figure 13. The scale is in decibels for the sound power level, in dimensionless variables. Thus a level of -100 dB corresponds to a dimensionless amplitude of 10 -5 for the PSE pressure fluctuation at the abscissa x = 0.1 D where the PSE solution starts. The amplitudes deduced from LES results on the conical surface, located outside of the jet, are first compared to the amplitudes deduced from LES data on a cylindrical surface R = 0.5 D, corresponding to the nozzle lip. These values are relatively close, for both m = 0 and m = 1, showing that the pressure fluctuations inside and outside of the jet can be represented by the same mode. At low frequency, the axisymmetric mode m = 0 has a larger amplitude than the first helical mode m = 1, but at higher frequencies both modes have a similar initial amplitude.

Considering the absolute amplitude deduced from the first POD mode of the linear antenna, it remains close to the values of the axisymmetric mode for frequencies up to St = 0.6 and slightly lower for higher frequencies. It is reminded that this value has been obtained without separation of the azimuthal modes and therefore it is more significant when only the axisymmetric mode dominates. Nevertheless, one order of magnitude on the amplitude of the pressure fluctuation corresponds to 20 dB on this graph and thus it can be stated that the LES is able to predict the correct order of magnitude of the pressure fluctuations. Moreover, this order of magnitude does not vary significantly with the frequency, in the considered range.

Conclusion

Based on three different data sets originating from numerical LES, semi-modal PSE approach and near-field pressure fluctuation measurements, a cross-comparison has been performed with the aim to get further insight on the existence and on the properties of large coherent structures in turbulent high-speed jets. Beside heavy numerical simulations like LES, the PSE approach is well suited to track the unstable modes developing in the jet shear layer. The evolution of these unstable modes directly predicted by the PSE exhibits strong similarities with results provided by the other approaches, giving a great presumption that they all describe the same physical phenomenon. However, before making this comparison possible, both LES and experimental results must undergo an appropriate modal decomposition along the three main coordinates, i.e. a Fourier decomposition in time and in azimuthal angle and a POD analysis in the axial direction. This analysis allows to isolate what seems to be the right mode corresponding to PSE results. This also confirms previous results published by several research teams. One key contribution of the present work is to confirm that the results obtained when performing the analysis at different radial positions remain coherent between LES and PSE. The radial structures of the PSE modes and of the modes extracted from LES are therefore very close. The PSE approach also allows to compare LES and experimental values when the data are not stored at the same position, which is the case in the present work.

Further investigations should however be done, like an improvement of the time convergence of the LES computation at greater axial positions, or an experimental investigation of the radial and azimuthal structures of the near-field pressure fluctuations. Using circular microphone arrays could give an insight on the azimuthal decomposition of the experimental pressure fluctuations. Moreover, recent works in progress in other laboratories take advantage of the Time-Resolved PIV to investigate the radial structure of the nearfield velocity fluctuations, which could be fruitfully compared to PSE solutions. Beside POD also exist other statistical techniques of modal decomposition, like Dynamical Mode Decomposition (DMD), which could be tested on the present numerical and experimental data. The question of understanding the physical nature of the fluctuations developing downstream of the potential core and represented by the higher-order axial POD modes, that is: everything except the basic Kelvin-Helmholtz mode, also requires a particular attention. Of course, the final objective of noise generation understanding will require far-field noise measurements together with far-field extrapolation of both LES and PSE solutions.

Figure 1 .

 1 Figure 1. Computed mean flow field and linear microphone array

Figure 2 .

 2 Figure 2. Axial velocity evolution along the jet axis

Figure 3 .

 3 Figure 3. PSE pressure fluctuation field -m = 0 -St = 0.4

Figure 4 .

 4 Figure 4. PSE pressure fluctuation amplitude for r = 0.5 D -m = 0 (left) and m = 1 (right)

Figure 6 .

 6 Figure 6. LES power spectral density for St = 0.4

Figure 9 .

 9 Figure 9. PSE modes amplitude adjustment on measurements -St = 0.3 (left) and St = 0.4 (right)

Figure 13 .

 13 Figure 13. Comparison of absolute amplitudes of the PSE modes fitted on LES or experimental results

Acknowledgements

The ORINOCO project was granted by the European Union, as part of the 7 th Framework Programme for Research and Technological Development, and by the Russian Government.

Appendix

The graphs representing the adjustment of the absolute amplitude of the PSE solutions on the LES values for the cylindrical surface R = 0.5 D, corresponding to the position of the nozzle lip and therefore to the centre of the shear layer, are presented in Figures 14 and15. These plots can be directly compared to those of Figures 7 and8 corresponding to the conical surface. In the case of the cylindrical surface, the decay occurs later than for the conical surface, where the maximum of fluctuation has a shorter axial extent. The initial amplitudes issued from both surfaces are compared in Figure 13.