Pagdame Tiebekabe

Ismaïla Diouf

NEW DIVISIBILITY TESTS

Keywords: Divisibility test, Congruence, Numerations, Algorithms

In this article, we first made the inventory of some various divisibility tests of 7 known, studied the effectiveness of these tests, demonstrate the new divisibility test of 7 proposed by the young Nigerian Chika Ofili and an algorithm in #Python. Secondary, we proposed and demonstrated a new test of divisibility by 3 which we called in this paper Diouf-Pagdame Test based mainly on the calculation of

where d and e represents respectively digits of tens and units of any given number. We have also emphasized the educational aspect of these methods.

Introduction

The divisibility of a number by 2 or 5 is very simple. It is known that a number N is divisible by 2 if its last digit is an even integer and is divisible by 5 if its last digit is 0 or 5. A number N is divisible by 3 if the sum of its digits is divisible by 3. However, these three tests can be easily explained using modulo arithmetic.

In this paper, we only deal with the case of the divisibility by 7 test and briefly discuss the 3-divisibility test. The 7-divisibility test has drawn the interest of some researchers. This gave birth to several techniques of divisibility. However, some tests have been proven more effective than others have, especially when it comes to verify the divisibility by 7 of a large number. In fact, if the divisibility of some numbers seems easy, it is however not the case of the test divisibility by [START_REF] Lagrange | Leçonns élém. sur les math. données à l'école normale en 1795[END_REF]. Some tests allow it to be done easily with few lines whereas for others, we need a lot of pages. It is in this thought that the recent discovery of a Nigerian youth Chika Ofili has caused a lot of ink in the newspapers, which had allowed him to win a British prize for the discovery of a new mathematical formula. Before, its discovery there were many existence tests. Among those tests, we have the Toja's test, the property of multiples of 7, the criterion for a large number, the method of utilization of a diagram, the criterion Pascal divisibility, and so on. These new tests of divisibility can be used for educational purpose due to their simplicity and efficiency. Therefore, we think we can start using them earlier in school environment.

In this article, we will list some known tests, demonstrate the divisibility test of 7 proposed by Chika, state and demonstrate a new divisibility test of 3 that we have discovered, compare these tests and determine the most effective of these tests by comparing the execution times of different algorithms in Python of each methods.

Historical Background

We can define a divisibility test as an algorithm, which uses the digits of an integer N to determine whether N is divisible by a divisor d. The history of divisibility tests started back from at least 500 C.E. when a divisibility test for 7 was included in the Babylonian Talmud. It was from then that methods which provide divisibility tests for all positive integers have been discovered and rediscovered by different mathematicians such as Joseph-Louis Lagrange [START_REF] Lagrange | Leçons élémentaires sur les math. données á l'école normale en 1795[END_REF], and Charles Dodgson (a.k.a. Lewis Carroll, author of Alice In Wonderland) [START_REF] Charles | Brief method of dividing a given number by 9 or 11[END_REF]. A remarkable summary of the literature in relation to divisibility tests published prior to 1915 is provided in Leonard Dickson's History of the New Divisibility Tests 3 Theory of Numbers [START_REF] Leonard | History of the Theory of Numbers[END_REF]. Edward Brooks dedicated two chapters to the study of divisibility tests in his 1880 book titled The Philosophy of Arithmetic [START_REF] Brooks | The Philosophy of Arithmetic[END_REF]. Recently in (2006), Marc Renault published a wonderful article [START_REF] Renault | Stupid divisibility tricks: 101 ways to stupefy your friends[END_REF] that provides divisibility tests for all integers between 2 and 102, which includes brief explanations for why the tests work. And lastly, Eric L. McDowell (Berry College) in Divisibility Tests: A History and User's Guide presents a modest survey of some of the more recent literature regarding divisibility tests.

Some Existing Divisibility Tests of 7

Property of Multiples of 7

There are already some effective methods that will allow to verify the divisibility of a given number. Among those tests, we can cite property of multiples of 7. The principle is quite simple. This test actually consists of verifying if the difference of the tens digit by the double of the units is a multiple of 7. If that difference is a multiple of 7, then the number to be tested is also a multiple of 7.

Criterion for a large Number

A second method based only on the fact that 103 is congruent to -1 modulo 7, is to separate this number in 3-digit slices starting from the units and insert alternately -and + between the slices. We carry out the operation thus written and this result is divisible by 7 if and only if the starting number was.

Example Let us take the number 5527579818992. We will separate it into three-digit sections starting from the units:

. 992 818 579 527 5

(1)

We interpose alternately -and +:

. 992 818 579 527 5

+ - + - (2)
We carry out the operation thus written:

. 231 992 818 579 527

5 = + - + - (3)
We

+ × + × + + × + × - …
Hence we can directly conclude that ( )

7 mod 0 ≡ N if ( ) .
7 mod 0 ≡ + - - + - a b x y z …

The Toja's Method

This third method is similar to the second one based only on the fact that

- + - = - - … (4) then (
). is a multiple of 7 (The different "remainders" are writtens in reverse order). We also notice that the divisibility by 7 of 1245545 is equal to that of 3136, and of 14, therefore 5527579818992 is divisible by 7.

Method of utilization of a diagram

This technique is based [START_REF] Brooks | The Philosophy of Arithmetic[END_REF] on the writing of a number in base 10 and on the modulo 7 congruence. The use of a diagram was proposed in 2009 by David Wilson [START_REF] Leonard | History of the Theory of Numbers[END_REF][START_REF] Talmud | [END_REF]. On a circle, we have all the numbers from 0 to 6, i.e. all possible modulo 7 remainders. We then match with an arrow each remainder r with the rest modulo 7 of . 10 * r

The diagram is then used as follows: for the integer:

• We stand on the case 0 and we are moving on the circle of n a cases.

We then obtain the remainder of an modulo 7;

• We then borrow the arrow, which starts from the square where we are and, from the point of arrival of the arrow, we move on the circle of 1 n a boxes. We then obtain the remainder of • We take the arrow which leads from 4 to 5. We move from 5 2 = a boxes and we arrive at 3;

• We take the arrow which leads from 3 to 2. We move from 6 1 = a boxes and we arrive at 1;

• We take the arrow which leads from 1 to 3. We move from 4 0 = a boxes and we reache 0.

The arrival box being 0, then the number 4564 is a multiple of 7.

Pascal's Criterion of divisibility

We divide 1 by 7, and we write at each stage of the calculation the successive remains. We obtain: This number is called Pascal's key.

With the number to be tested, we multiply the number of units by 1, the tens digit by 3, the hundreds by 2, and so on. Finally, we add the results obtained. If the number obtained is a multiple of 7, then the initial number to be tested is a multiple of 7.

Example 2.5. 644 is a multiple of 7, because 28 2 6 3 4

1 4 = × + × + ×
which is a multiple of 7. So therefore, 644 is also a multiple of 7.

Proof of Pascal's criterion

If we perform the division of 1 by 7, we obtain 0; 142857 and a sequence of remains in the division: Let N a number of n digits such that 1 1 10 10

… , 5 ,
- -× + × = n n n n N N N 0 1 1 10 N N + × + + … with [ ] 9 , 0 ∈ i N and . N ∈ i N
By applying the calculation suggested by the criterion, we obtain

( ) ( ) ( ) ( ) 3 1 7 7 3 7 1 0 7 1 1 0 1 0 × + × + = + × + + + × + + × N N N r d N N N n n n … . 7 n n n n r N d N + + + …
The first terms of each parenthesis being multiples of 7, we can ignore them. If

0 1 1 1 1 10 10 10 N N N N N n n n n + × + + × + × = - - … is a multiple of 7, then ( ) ∑ + i i i r d N 7
(which by hypothesis is equal to the previous expression must be divisible by 7 for little ∑ i i r m so are.

New Divisibility Tests

Statement

Proof of Theorems

We will prove the two theorems together. Let m be the number to be tested. m is written in the form of e d m + = 10

where e is the last digit and d is the remainder of the number. m can also be written in the form

+ = d m 10 ( ) (
). This ends the proof.

Main Results and Observations

Main results

In this section, we present the execution time of the different programs on ASUS CORE i5 8th Gen. For a sufficiently given large number, we evaluate the Python code of the various tests listed in this article, and we note the execution time in a summary tables. We have chosen two numbers, one of which is divisible by 7 and the other is not. This in order to verify the effectiveness of the various tests. These numbers are Below are the tables of the execution times of the different methods.

Remarks

By the help of Python programs on the different tests, we can compare Pagdame Tiebekabe and Ismaïla Diouf 10 them by using the execution time as a parameter of comparison. Indeed, we randomly chose a number large enough (of the order of 60 digits) to test the programs. As the tables above indicate, the differences appear to be negligible. however small it is, makes all the difference between the different methods. We have kept several digits after the comma to be as precise as possible. The difference in execution time between Method 5 (the most efficient) with the lowest execution time and Method 2 (the least efficient) with the longest time is 0:004250008 and 0:006796797 from Table 1 and Table 2 respectively. In view of these results, taking the execution time as a comparison parameter, method 5 seems to be the most efficient and method 2 is the least efficient among those methods which have been studied.

Conclusion

Finally, we will conclude that these methods are all very efficient. However, some are more efficience than others in their simplicity. Simple in their understanding and execution. According to the execution times, Pascal's criterion is the most efficient among the tests of divisibility by 7. Pascal's criterion is most efficient in view of the results obtained in the section 6 but is little too heavy. It is based on interesting property which emerges in the passage from base 7 to decimal base. It uses a sequential digit key. But its inconvenience side is usefulness. It is advantageous when we deal with several numbers. Hence, we should highlight many similarities between the tests. Between the method of property of multiples of 7 and Diouf-Pagdame test, we can notice the use of the number u d 2 in the two cases. The methods "criterion of large number" and the "Toja's method" have in common the separation by slices from the units. The first by the slice of 3 and the second by the slice of 2.

3 10

 3 is congruent to -1 modulo 7, from which we deduct that if si

•aExample 2 . 4 .

 24 We then repeat the process (borrowing an arrow, and then moving on the circle) until . 0 Pagdame Tiebekabe and Ismaïla Diouf 6The number is divisible by 7 if and only if the arrival box is box 0.This figure is:Standard and we move on the boxes according to the procedure stated previously. Suppose we want to verify the divisibility by 7 of 4564.

1 :Method 2 :

 12 Property of multiple of 7; Criterion for a large number; Method 3: Toja's method; Method 4: Method of utilization of diagram and Method 5: Pascal's method.

  Any integer N can be written in the form:

						see if 231 is divisible using the divisibility lemma by
	7	:	23	+	5	*	1	=	28	is divisible by 7 so 5527579818992 is.
			Proof. Since	( 1 1000 -≡	mod	7	)	and	1000000 ≡	( 1	mod	7	),	then for all
	integer	n	>	, 0	we have: ( ) 10 2 3	n	≡	( 1	mod	7	)	and ( ) 10 2 3	n	-	1	≡	-	( 1	mod	7	).
	a		( ) 10 2 3	n			b	( ) 10 2 3	n	1						( ) 10 3	y	10	3	. z

  From the right, the nearest multiple of 7 by default is 91: the distance

	92	-	91	=	; 1
		• For the second pair, the nearest multiple of 7 by excess is 91: the
	distance	91	-	89	=	; 2
		• For the third pair, the nearest multiple of 7 by default is 77: the
	distance	81	-	77	=	; 4
		• For the fourth pair, the distance:	84	-	79	=	, 5	and so on.
		The starting number is a multiple of 7 if and only if
	1	2	4		5		5	4	5
									10	x m ≡	y	mod	7	(5)
		Example 2.3. Let us take the number 5527579818992. We will separate
	it		into			two-digit	sections	starting	from	the	units
	5	52	75	79	81	89	. 92

Therefore, x is divisible by 7 if and only if y is. We can definitely replace by the way every i b by any integer, which is congruent to modulo 7. The principle is therefore to cut the number x into slices of 2 digits and find the distance between each number of 2 digits and the nearest multiple of 7 (alternately by excess and by default).

•

  Here we give a proposal of a new theorem.

										of Chika's divisibility test
			Let d, e be non-negative integers,	e	≤	9	and	m	=	10	d	+	. e	Thus
	7	|	m	⇔	7	|	d	+	. e 5
			Theorem (Diouf-Pagdame). Let d, e be non-negative integers,	e	≤	9
	and	m	=	10	d	+	. e	Then	3	|	m	⇔	3	|	d	-	2	. e

Table 1 .

 1 For =

	1 N	923456778909811325377849865278365475778276485
	7653512536123456789098760774
	Methods	Method 1	Method 2	Method 3
	Time of executions 0:024769759000264457 0:024846616999866455 0:024533451000024797
		Method 4	Method 5
		0:022657221000145 0:020596608999994714

Table 2 .

 2 For =

	N	2	34526780209364787384690124356272134567890272
	134563767471234567897234272345
	Methods		Method 1	Method 2	Method 3
	Time of executions 0:022463982999852306 0:023879767999915202	0:02267362900010994
			Method 4	Method 5
			0:023988764000023366	0:017082971000036196


Pagdame Tiebekabe

Ismaïla Diouf

POWERS OF THREE AS DIFFERENCE OF TWO FIBONACCI NUMBERS ( ) p m n F F 3 =

Keywords: linear forms in logarithm, diophantine equations, fibonacci sequence, Lucas sequence

The goal of this paper is to find, in a simple and rigorous way, all powers of three as the difference of two Fibonacci numbers, that is, we study a diophantine equation

n, m, and p where ( ) 0

is the Fibonacci sequence. The tools used to solve our main result are properties of continued fractions, linear forms in logarithms, and a version of the Baker-Davenport reduction method in diophantine approximation.

Introduction

The Fibonacci and Lucas numbers 1, 1, 2, 3, 5, 8, … and 2, 1, 3, 4, 7, 11, … respectively have fascinated mathematicians for over hundred years. respectively and belong to a list of particular member of a general family of sequences. The determination of perfect powers of Lucas and Fibonacci sequences does not date from today. They have many interesting properties and have been studied by many researchers. For a brief history of Fibonacci and Lucas sequences one can consult [START_REF] Debnath | A short history of the Fibonacci and golden numbers with their applications[END_REF]. Proprieties of Fibonacci and Lucas sequences, and the relationship between them, have been the focus of a considerable amount of research. The real contribution of determination of perfect powers of Lucas and Fibonacci sequences began in 2006. By classical and modular approaches of Diophantine equations, Bugeaud et al. [START_REF] Bugeaud | Fibonacci numbers at most one away from a perfect power[END_REF] defined all perfect powers of Lucas and Fibonacci sequences by solving the equations Motivated by the studies of Bravo and Luca [1, 2], our main focus is on the powers of 3 as a difference of two Fibonacci numbers. We consider the Diophantine equation

p m n F F 3 = - (1)
in nonnegative integers

Auxiliary Results

In this section, we recall some important results used by us. The wellknown Binet formulas provide closed form rules to calculate the Fibonacci and Lucas numbers, both incorporate the so-called golden ratio ( ) , 2 5 1 + hinting at a deep connection between the sequences:

5 n n n F β - α = and , n n n L β + α = where ( ) 2 5 1 + = α and
( ) 2 5 1 - = β
are the roots of the characteristic

equation . 0 1 2 = - -x x
The relation between Fibonacci and Lucas numbers and α are:

, 1 n n n L F F = + + (2) , 2 2 - - α ≤ ≤ α n n n F for . 1 ≥ n ( 3)
(3) can be proved easily by induction, and together with (2) allows us, in Section 3, to prove the following important inequality: .

1 n p < ≤ It can be seen that 2 1 < α < and . 0 1 < β < -
The following theorem due to Lucas and Patel is given in [4].

Pagdame Tiebekabe and Ismaïla Diouf 4 Theorem 2. All solutions of Diophantine equation

,

p m n y F F = ± 2 ≥ p in integers ( ) p y m n , , , with
( ) 2 mod m n ≡ have either { } 36 , max < m n or 0 = y and . m n =
Theorem 2 is a generalization of the following theorem due to Bugeaud et al. [START_REF] Bugeaud | Fibonacci numbers at most one away from a perfect power[END_REF].

Theorem 3. The only nonnegative solutions ( )

p y n , , of the equation p n y F = -1 with 2 ≥ p are . 2 1 , 1 1 , 0 1 1 2 5 3 2 1 = - = - = - = - F F F F
We also need the following theorem due to Matveev and Lemma due to Dujella and Pethö [START_REF] Matveev | An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II[END_REF][START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF].

Theorem 4 (Matveev [6]

1 l b b B = and . 1 1 : 1 1 1 - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ η = - η η = Λ ∏ = l i b i b l b i l Let l A A ..., , 1 reals numbers such that { ( ) ( ) } . 1 , 16 . 0 , log , max l j Dh A j j j ≤ ≤ η η ≥ Assume that , 0 ≠ Λ then we have ( ) ( )( ). log 1 log 1 1 30 3 log 1 2 5 . 5 4 nB D A A d l l l + + × × + × × - > Λ + Further, if L is real, then ( ) ( )( ). log 1 log 1 30 4 . 1 log 1 2 5 . 4 3 B D A A d l l l + + × × × × - > Λ +
During our calculations, we get upper bounds on our variables which are too large, so we have to reduce them. To do this, we use some results from the theory of continued fractions. In particular, for a non-homogeneous linear form with two integer variables, we use a slight variation of a result due to Dujella and Pethö [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF]. For a real number X, we write = :

X { } Z ∈ - n n X : min
for the distance of X to the nearest integer.

Lemma 5 (Dujella and Pethö [7]). Let M a positive integer, let q p be the convergent of the continued fraction of the irrational number γ such that M q 6 > and let μ , , B A be real numbers such that 0

> A and . 1 > B Let . : q M q γ - μ = ε If 0 > ε , then the inequality , 0 k AB n m - < μ + - γ <
does not admit a solution in integers m, n and k such that M m ≤ and

( ) . log log B Aq k ε ≥
The following properties of the logarithmic height is used in the next section:

• ( ) ( ) ( ) . 2 log + η + γ ≤ η ± γ h h h • ( ) ( ) ( ). 1 η + γ ≤ γη ± h h h • ( ) ( ). γ = γ h s h s

Main Result

In this section, we give the proof of Theorem 1. The only solutions ( )

p m n , , of the Diophantine equation p m n F F 3 = - in nonnegative integers n m < and p are ( ) ( ) ( ) ( ) ( ), 1 , 0 , 4 , 0 , 2 , 3 , 0 , 1 , 3 , 0 , 0 , 2 , 0 , 0 , 1 ( ) ( ) ( ) ( ).
( ) ( ) ( ) { }. 0 , 2 , 3 , 0 , 1 , 3 , , ∈ p m n Now assume that { }. 2 , 1 ∈ -m n Then (1) becomes 2 3 - = n p F for 1 - = n m or 1 3 - = n p F for , 2 - = n m according to [4], ( ) = p m n , , ( ) ( ) ( ) ( ) ( ) ( )
≥ = - ≥ - = + + m m m m n p F F F F F (6)
using (2). From ( 5) and ( 6 We now assume that n and m have different parities.

. 2

1 5 5 3 5 3 + α < β + α < β + = - α ⇒ = - m n m n m p n p m n F F F Dividing both sides, by , 5 n α we get . 4 5 3 1 m n n p - - α < α - (7)
Now applying Matveev's theorem with the following data: , : , : , 5 : , : , 3 :

2 1 3 2 1 n b p b - = = = γ α = γ = γ and . 1 : 3 = b Since ( ). 5 , , 3 2 1 Q ∈ γ γ γ
Then we can take ( ).

5 Q K = So, ( ) . 2 5 deg 2 = - = x D 1 5 3 1 - α = Λ -n p is nonzero. Otherwise, we get p m n n n n p p F F F F 3 1 5 5 3 = - > - > β + = α =
which is impossible.

or we can prove it is nonzero by using the conjugate of the relation . 5 3

p p α = Moreover ( ) ( ) ( ) . 5 log , log 2 1 , 3 log 3 2 1 = γ α = γ = γ h h h Then we can take, , 5 . 0 , 2 . 2 2 1 = = A A and . 7 . 1 3 = A Since , n p < it follows that { } . 1 , , max : n n p B = - =
Since all conditions are satisfied, then from Theorem 4, we have (  )

1 4 Λ > α -m n ( ( ) ( ) ) 7 . 1 5 . 0 2 . 2 log 1 2 log 1 2 3 30 4 . 1 exp 2 5 . 4 6 × × × + + × × × - > n hence ( ) 4 log log - α -m n ( )( ) 7 . 1 5 . 0 2 . 2 log 1 2 log 1 2 3 30 4 . 1 2 5 . 4 6 × × × + + × × × <
3 1 5 3 5 1 < β + β ≤ - α - α - m n p n m n hence ( ) n n n m p α < α α - - - - - 3 1 5 3 1 1 (9) because ( ) . 3 1 3 1 1 3 2 1 1 1 < α - ⇒ > α - ⇒ < α < α = α - - - - - n m n m m n n m
Q ∈ γ γ γ
Then we can take ( ).

5 Q K = So, ( ) . 2 5 deg 2 = - = x D ( ) . 1 1 5 3 : 1 2 - α α - = Λ - - - n n m p Let prove that 0 2 ≠ Λ by contradiction. If , 0 2 = Λ then we have 5 5 5 3 m n m n p β = β ⇒ α - α = which is impossible since . m n > Similarly ( ) ( ) ( ) ( ) α - + ≤ γ α = γ = γ log 2 1 5 2 log , log 2 1 , 3 log 3 2 1 m n h h h × + × × × + + <
- ≤ - < < 80 1 1 0 1 1 1 1 or m n n p - α < + α - < 80 5 log log 3 log 0 Then 2 2 1 3 1 1 2 2 2 < ⇒ < α < - = - z n z z e e e
and therefore

n n z z z z e e e z α < < ⇒ α < - = - < < 6 0 6 1 1 0 2 2 2 2 2 this is equivalent to ( ( ) ) n n m n p α < α - + α - < - - 6

Fibonacci

  terms and later perfect powers in the Fibonacci and Lucas sequences have attracted the attention of the researchers. From there, many researchers tackled similar problems. It is important to recall that, in Corollary 1 of [1], Kebli et al. give the condition on all integer solutions of equation . The proof of their Theorem 2 is based on abc-conjecture (open problem). But it also uses an upper bound on m nwhich appears in the proof of Theorem 1 of [1] and rests on an application of lower bounds for linear forms in logarithms of algebraic numbers. One can consult [1] for more information. Our results, respect all conditions given in [1].

From

  S. Kebli, O. Kihel, J. Larone and F. Luca, On the nonnegative integer solutions to the equation , Theory 220 (2021), 107-127. [2] J. J. Bravo and F. Luca, On the Diophantine equation , Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), 969-1018. [4] F. Luca and V. Patel, On perfect powers that are sums of two Fibonacci numbers, J. Number Theory 189 (2018), 90-96.

  m, n, and p. We follow the approach and the method

						Powers of Three as Difference of Two Fibonacci …	3
	a version of the Baker-Davenport reduction method in diophantine
	approximation.
		This paper follows the following steps: In Section 2, we introduce
	necessary theorems and lemma used in Section 3 to prove the main result of
	this paper stated below:
		Theorem 1. The only solutions (	m n , ,	p	)	of the Diophantine equation
	n F	-	m F	=	3	p	in nonnegative integers	n m < and p are
					( 1 ,	, 0	0	) ( ,	, 2	, 0	0	) ( 3 , ,	, 1	0	) ( 3 , ,	, 2	0	) ( ,	, 4	, 0	), 1
					(	, 4	, 3	0	) ( 5 , ,	, 3	) ( , 1	, 6	, 5	) ( 11 , , 1	, 6	4	).
	presented By Luca and Bravo in [2]. The proof of our main result uses lower
	bounds for linear forms in logarithms, properties of continued fractions, and

  .

	, 2	, 0	0	,	, 3	, 1	0	,	, 3	, 2	0	,		, 4	, 3	0	,	, 5	, 3	1	,	, 6	, 5	1	With the aid of
	Mathematica, we obtain the solutions in the statement for	1	≤	m	<	n	≤	200	.
		For now, assume		n		>	, 200	m		≥	3	and	n	-m	≥	. 3	Let us prove an
	inequality which is important for our purpose. From	α	n	-	2	≤	n F	≤	α	n	1 -	,	we
	get
						3	p	=	n F	-	m F	<	n F	<	α	n	<	n 3	i.e.,	3	p <	3	. n	(5)
		Since	m	>	3	and	n		-m		≥	, 3	it follows that
							3													3			2	1	6

and so we take ( )

), Two cases arise: when 0 2 > z and . 0 2 < z


Romain Gojon

Ephraim Gutmark

Mihai Mihaescu

On the response of a rectangular supersonic jet to a near-field located parallel flat plate

In this paper, the flow and acoustic fields of a rectangular over-expanded supersonic jet interacting with a parallel plate are investigated using compressible LES. The jet exits from a converging diverging rectangular nozzle of aspect ratio 2 with a design Mach number 1.5. Four simulations with four different distances between the lower inner lip of the rectangular jet in the minor axis plane and the plate ranging from 0 to 3 equivalent diameters are performed. The geometry of the nozzle, the positions of the plate, and the exit conditions are chosen in order to match those in an experimental study conducted at the University of Cincinnati. Snapshots and mean velocity fields are first presented. A good agreement with the PIV experimental measurements is found. The overall sound pressure levels are then plotted along the minor and major axis. In a previous paper, the corresponding free jet has been found to undergo a strong flapping motion in the minor axis plane due to the screech feedback mechanism. In the present study, it is seen that the intensity of the screech feedback mechanism increases for some distances from the ground and decreases for some others, as compared to the corresponding free jet. A study of the jets shearlayers is then proposed, first by looking at two points space-time cross correlations of the axial velocity. The convection of the turbulent flow structures is thus studied. Then, two points space-time cross correlations of the pressure along the jets shear-layers are proposed and an amplification of the aeroacoustic feedback mechanism leading to screech noise is observed in the lower jet shear-layers for two cases. It is also observed that the screech feedback mechanism establishes mainly between the nozzle lips and the end of the tenth shock cell. The acoustic loading on the plate is finally studied. As pointed out in a previous study, the flapping motion of the jet at the screech frequency seems to yield to an asymmetric organization of the Mach wave radiation also at the screech frequency. Those organized Mach waves impinge on the plate, are reflected, and propagate back towards the jet, exciting the shear-layer at the screech frequency. This will amplify the screech mechanism in the lower jet shear-layer. However, this amplification happens only for some nozzle-to-plate distances. The mechanism leading to this behaviour is explained.

I. Introduction

Rectangular propulsion systems closely integrated to the fuselage are considered for future high-speed aircrafts. The jet plume can thus interact with nearby parallel surfaces, from the aircraft during cruise or from the ground during take-off.

For subsonic rectangular jets, several experimental [START_REF] Yu | Experimental investigation of the trailing edge noise mechanism[END_REF][START_REF] Bridges | Noise from aft deck exhaust nozzles -differences in experimental embodiments[END_REF][START_REF] Zaman | Resonant interaction of a rectangular jet with a flat-plate[END_REF] studies have been conducted in order to study the interactions between the jet flow and a parallel plate and the consequences on the noise generation mechanisms. Notably, for some jet Mach numbers, aspect ratios, and plate locations, an acoustic tone can be generated. This has been observed experimentally by several researchers [START_REF] Bridges | Noise from aft deck exhaust nozzles -differences in experimental embodiments[END_REF][START_REF] Zaman | Resonant interaction of a rectangular jet with a flat-plate[END_REF] . For supersonic rectangular jets, very few studies have been conducted [START_REF] Mora | Acoustics from a rectangular supersonic nozzle exhausting over a flat surface[END_REF] . However, several studies have been conducted for supersonic round jets interacting with a parallel plate [START_REF] Mclaughlin | Experiments on the effect of ground reflections on supersonic jet noise[END_REF][START_REF] Brown | Investigation of broadband shock noise from a jet near a planar surface[END_REF][START_REF] Liu | Impact of deck and jet blast deflector on the flow and acoustic properties of an imperfectly expanded supersonic jet[END_REF][START_REF] Powers | Noise reduction with fluidic inserts in supersonic jets exhausting over a simulated aircraft carrier deck[END_REF] . For round jets, the effects of the plate on the shock cell structure have been observed to be minor, except when the plate is very near to the nozzle and interacts with the flow field [START_REF] Brown | Investigation of broadband shock noise from a jet near a planar surface[END_REF][START_REF] Liu | Impact of deck and jet blast deflector on the flow and acoustic properties of an imperfectly expanded supersonic jet[END_REF] . Moreover, in terms of acoustic radiation, it has been shown [START_REF] Mclaughlin | Experiments on the effect of ground reflections on supersonic jet noise[END_REF][START_REF] Powers | Noise reduction with fluidic inserts in supersonic jets exhausting over a simulated aircraft carrier deck[END_REF] that the reflections from the flat plate appear as constructive and destructive interferences in the sideline spectrum. Finally, Mora et al. [START_REF] Mora | Acoustics from a rectangular supersonic nozzle exhausting over a flat surface[END_REF] recently observed, for supersonic rectangular screeching jet interacting with a flat plate, an amplification of the feedback mechanism leading to screech noise for specific positions of the plate.

In the present paper, compressible jets exiting from a rectangular converging-diverging nozzle and interacting with a parallel plate are simulated in order to study the influence of the plate on the flow and on the acoustic field. An experimental study of a similar jet is conducted at the University of Cincinnati 4 . In the experiments and in the present study, an amplification of the aeroacoustic feedback mechanism leading to screech noise is observed for some plate positions. An explanation of the resonant interaction between the jet and the flat plate, leading to this amplification, is attempted. The paper is organized in seven sections. The jet parameters and the numerical parameters are first given in section II. Sections III and IV present the aerodynamic and acoustic results, respectively. Notably, hydrodynamic properties are compared with experimental data. The organization of the jet shear-layers is analyzed in section V and the acoustic loading on the plate is studied in section VI. Finally, concluding remarks are given in section VII.

II. Parameters of the study

II.A. Jet parameters

Large eddy simulations are performed for four rectangular jets corresponding to four locations above a plate, between 0 and 3D eq from the inner lip in the minor axis plane where D eq is the diameter of an equivalent round jet. The different cases are presented in table 1. The jets will be referred as JetL0, JetL1, JetTL2 and JetTL3 in the following. They originate from a rectangular converging-diverging nozzle of height h and aspect ratio 2 with a design Nozzle Pressure Ratio 3.67. They are over-expanded with a Nozzle Pressure Ratio (NPR) of 3 and they have a temperature ratio (TR) of 2. The geometry, the exit conditions and the locations of the plate are similar to those in the experimental study of Mora et al. [START_REF] Mora | Acoustics from a rectangular supersonic nozzle exhausting over a flat surface[END_REF] . It is also worth noting that the corresponding free jet has also been computed and analyzed in a previous study by Gojon et al. [START_REF] Gojon | Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet[END_REF] Some results from this free jet will be used for comparison purposes.

L

N P R M j T R u j m.s In order to better apprehend the problem, a 3-D visualization of the nozzle, the plate and the jet is proposed in Figures 1 and2 and for JetL2 and JetL3, respectively. It is worth noting that for JetL1 and JetL2, the plate is modified under the nozzle, having a design that avoids possible issues related with air entrainment from upstream. The constructed valleys are not exactly similar to the geometry in the experiments and small differences can thus be expected for those two cases.

II.B. Numerical parameters

In this numerical study, the aerodynamic fields as well as the near acoustic fields are computed in order to understand the effects of the plate on the noise generated by the jet. To reach this objective, structured grids consisting of 64 blocks and containing about 155 millions of nodes are used. They have been designed in order to allow acoustic waves with Strouhal numbers up to St = f D eq /u j = 3 to be well propagated, where f is the frequency. The simulations are using a compressible flow solver 10 already assessed in a previous study [START_REF] Gojon | Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet[END_REF] , running on a supercomputer on 960 processors. In particular, an explicit four-stage Runge-Kutta algorithm is used for time integration and a central discretization of second order of accuracy is used for spatial discretization. Moreover, at the end of each time step, a Jameson type artificial dissipation is added to the inviscid flows in order to remove grid-to-grid oscillations, to avoid Gibbs oscillations near shocks and to relax subgrid-scale turbulent energy. This artificial dissipation is derived from the one proposed by Ducros et al. [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF] The exact formulation can be found in Gojon et al. [START_REF] Gojon | Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet[END_REF] A total of 300, 000 time-steps are computed in each case after the transient period, permitting to have a simulated time of 600D eq /u j .

III. Aerodynamics results

III.A. Snapshots

Snapshots along the minor and major axes planes are displayed in Figure 3, permitting to visualize simultaneously the jet flow and the acoustic field.

For the case JetL0, with a plate directly aligned with the inner lip in the minor axis plane, the presence of the plate alters the shock cell structure of the jet and the near acoustic field. Notably, only few shock cells are visible in the hydrodynamic field and the strength of the shocks is lower compared to the three other considered jets. In the acoustic field, three main components can be seen. First, Mach waves are visible in the major and minor axis planes. This specific noise component appears when the convection velocity of the turbulent structures in the jet shear-layers becomes supersonic. This acoustic component has been studied experimentally [START_REF] Panda | Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise[END_REF] and numerically [START_REF] Nichols | Large eddy simulation of crackle noise in hot supersonic jets[END_REF] , it has a marked directivity and a broadband spectrum in general. The second acoustic component visible propagates in the downstream direction. It is associated to a low frequency. This acoustic contribution corresponds to the mixing noise, as observed by Bogey and Bailly 14 , Sandham and Salgado 15 and Tam [START_REF] Tam | Mach wave radiation from high-speed jets[END_REF] for instance. Finally, circular acoustic waves can be seen exiting from the jet shear layers at different axial positions. Those acoustic waves are created by the interactions between the shock cell structure and the turbulent structures in the jet shear-layers. In the far-field, those waves lead to the observation of broadband shock-associated noise [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF][START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF][START_REF] Dahl | Predictions of supersonic jet mixing and shock-associated noise compared with measured far-field data[END_REF] . For the three other jets, shock cell structures containing almost ten cells are visible in the hydrodynamic fields. In the acoustic fields, the three former components can be seen along with a fourth one. Indeed, acoustic waves propagating in the upstream direction in the vicinity of the nozzle are visible in the minor axis plane. These waves are linked to the screech noise mechanism. This noise component have been observed in various studies, notably in Westley and Woolley 20 , Tam and Tanna 17 , Panda et al. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] , and André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] These waves are organized antisymmetrically with respect to the jet axis. This suggests a screech mechanism associated with a flapping motion of the jets in the minor axis plane. Those qualitative results are in agreement with the results of the experimental team [START_REF] Mora | Acoustics from a rectangular supersonic nozzle exhausting over a flat surface[END_REF] . They observe the suppression of the screech noise component and a reduced number of shock cells for the case with a plate directly aligned with the inner lip in the minor axis plane. For the three other cases, a flapping motion of the jet in the minor axis plane was also observed.

III.B. Mean flow fields

For the four present jets, the mean axial velocity distributions are visible in Figure 4 in the minor and major axis planes, respectively. For JetL0, in Figure 4(a,e), only a couple of shock cells are visible downstream of the nozzle exit. For the three other cases, the shock cell structures are similar, with around ten shock cells visible in the jet plume. The same evolution can be seen in the experimental PIV results, also plotted in Figure 4. Moreover, those results are consistent with several experimental studies showing that a plate located in a parallel way to a supersonic jet has very little influence on the shock cell structure except if the plate is very near to the nozzle and interacts with the flow field [START_REF] Brown | Investigation of broadband shock noise from a jet near a planar surface[END_REF][START_REF] Liu | Impact of deck and jet blast deflector on the flow and acoustic properties of an imperfectly expanded supersonic jet[END_REF] .

III.C. Overall sound pressure levels

Overall sound pressure levels in the near field are computed using the root mean square pressure fields. A characterization of the effect of the plate on the overall acoustic radiation of the jets is thus possible. Overall Sound Pressure Levels (OASPL) in the minor and major axis planes are represented for the four jets in Figure 5. Isocontours are added in order to improve the readability.

Two main acoustic components can be distinguished. The first acoustic component is linked to acoustic waves propagating in the upstream direction, permitting to close the aeroacoustic feedback loop leading to screech noise. This acoustic component is only visible along the minor axis and its amplitude is very low for JetL0 in Figure 5(a) and higher with values about 190 dB for JetL1, JetL2 and JetL3, in Figures 5(b,c,d).

The second acoustic components visible is linked to the Mach wave noise. This acoustic component is visible in both the minor and major axis planes, as already observed qualitatively in Figure 3. By looking at the snapshots in Figure 3 and at overall sound pressure levels in Figures 5,a angle of propagation of about 40 degrees with respect to the jet axis is found. This result is in agreement with the model of Oertel 23 using the convective Mach number M c = 1.25 found previously [START_REF] Gojon | Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet[END_REF] for the corresponding free jet. The two other acoustic components observed in the snapshots, namely the mixing noise an the broadband shock-associated noise, cannot be distinguished.

IV. Acoustic results

IV.A. Near-field acoustic

For the four jets, acoustic spectra at (x, y, z) = (0, h, 0) are plotted in Figure 6. The acoustic spectrum of the corresponding free jet is added in lightgrey. For this free jet, a screech tone frequency has been found at a Strouhal number of St = 0.315 and a low frequency was observed at St = 0.18 [START_REF] Gojon | Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet[END_REF] . For JetL0, in Figure 6(a), the presence of the plate changes completely the acoustic spectrum. A low frequency is now observed at St = 0.13 and a screech frequency is observed at St = 0.32 but with an amplitude 18 dB lower as compared to the one of the free jet. For the other jets (JetL1, JetL2, JetL3), the low frequency and the screech frequency of the free jet are visible. However, it is worth noting that the first harmonic of the screech frequency, visible in the spectrum of the free jet at St = 0.63, cannot be seen in the spectra of the four present jets. The frequencies and the amplitudes of the low frequency and of the screech frequency have been gathered in Table 2. The low frequency increases with increasing the distance from the plate, from St = 0.13 for JetL0 to St = 0.19 for JetL3. For the last case, a value similar to the one of the corresponding free jet is found. The screech frequency remains the same for all the cases, at St ∼ 0.32, a value similar to the one of the corresponding free jet. However, the amplitude of the screech frequency changes with the location of the plate. For JetL0, the amplitude of the screech frequency is very low compared to the one of the free jet. For JetL1 and JetL2, it is very interesting to note that the amplitude of the screech frequency is higher than the one of the corresponding free jet. Finally, for JetL3, the amplitude of the screech frequency is 4.5 dB lower than the one in the corresponding free jet. A resonance mechanism between the jet and the plate, increasing the amplitude of the screech frequency, is thus observed for JetL1 and JetL2. Similar observations have been made in the experimental study of Mora et al.

V. Organization of the jets shear-layer

The jet shear-layers are studied by computing cross correlations of axial velocity and of fluctuating pressure. Differences in the developments of the jet shear-layers, between the upper and the lower ones, are discussed.

V.A. Cross correlations of axial velocity

The hydrodynamic properties of the jet shear-layers are studied by computing two point space-time cross correlations along the lower and upper jet shear-layers in the minor axis plane. In order to achieve this goal, 150 monitoring points are used along the lower and upper jet shear-layers, at y = ±h/2 and for 0 < z < 15h. For each shear-layer, a reference point is chosen at z = 7.5h and the cross correlation is computed with a second point ranging from z = 0 to z = 15h. For the axial velocity for example, the space-time normalized cross correlation is computed with the relation

R(z, τ ) = < u(x ref , y ref , z ref , t ref )u(x ref , y ref , z, t ref + τ ) > < u(x ref , y ref , z ref , t ref ) > 2 < u(x ref , y ref , z, t ref + τ ) > 2
where u is the axial velocity, x ref = 0, y ref = h/2 for the upper shear-layer and y ref = -h/2 for the lower shear-layer, z ref = 7.5h, z ranges from 0 to 15h, t ref is a reference time and τ is the separation time.

The space-time cross correlation computed with the above formula are plotted in Figures 7 and8 in the minor axis plane for the lower and the upper shear-layers, respectively. A clear pattern is visible with an inclined line of strong cross correlation in red. This strong cross correlation is due to the downstream propagation of turbulent structures in the shear-layers. The angle of this line at the reference point (i.e. at (z, τ ) = (7.5h, 0)) corresponds to the local convection velocity of the turbulent structures in the shear-layers at z = 7.5h.

The same cross correlation pattern can be computed with different reference points ranging from 0 to 15h in order to find the local convection velocity of the flow structures as a function of the axial coordinate. The results for the upper and the lower shear-layers are given in Figure 9 where the local convection velocity is plotted as a function of the axial coordinate. The results for the corresponding free jet are also represented. For JetL1, JetL2 and JetL3, the convection velocity is not constant but varies according to the shockcell structure, as in the free jet. The same observations have already been made experimentally [START_REF] André | Etude expérimentale de l'effet du vol sur le bruit de choc de jets supersoniques sous-détendus[END_REF] and numerically [START_REF] Gojon | Flow structure oscillations and tone production in underexpanded impinging round jets[END_REF] . Moreover, for JetL1, JetL2 and JetL3, the local convection velocity is about 10% lower than the one in the corresponding free jet and the convection velocity along the lower shear-layer is slightly larger than the one along the upper one. Finally, for JetL0, the decrease of the strength of the shock cell structure leads to a convection velocity almost constant in the upper shear-layer along the axial direction. In the lower shear-layer, the patterns were not clear enough to find the local convection velocity.

V.B. Cross correlations of pressure

In order to be able to observe both hydrodynamic and acoustic waves, the same space-time cross correlation is applied using the fluctuating pressure. The results for the lower and upper jet shear-layers in the minor axis plane are given in Figures 10 and11, respectively. In the jet shear-layer, the fluctuating pressure field is a superimposition of the hydrodynamic pressure and of the acoustic pressure. On one hand, the hydrodynamic component is supposed to be similar to the component observed in the cross correlation of axial velocity. On the other hand, the acoustic pressure comes from the acoustic waves propagating in the upstream direction observed in Figure 3 and linked to the screech noise component. In order to better understand, the results of the lower shear-layer of JetL1, represented in Figure 10(b), are considered first. Different stripes organized vertically can be seen, indicating a strong temporal periodicity of the fluctuating pressure. This periodicity is due to the screech mechanism, organizing the fluctuating pressure in the jet shear-layers. Indeed, a temporal periodicity of T = 1.15 × 10 -4 s is found in Figure 10(b). This temporal periodicity is linked to a Strouhal number St = D eq /(T u j ) = 0.32 which is identical to the screech frequency observed in Figure 6(b). Moreover, this temporal periodicity is the strongest for JetL1 and JetL2 in Figure 10(b,c), is still visible in Figure 10(d) for JetL3 and is not visible anymore for JetL0 in Figure 10(a). This observation is consistent with the amplitudes of the screech frequency reported in Table 2. Finally, for JetL1, JetL2 and JetL3, in Figure 10(b,c,d), two main regions can be seen as a function of the axial direction. For 3h < z < 12h, a region with lines is visible. The slope of the lines is positive, indicating instabilities propagating in the upstream direction. This result means that in this region, the pressure is mainly driven by upstream propagating instabilities. If those instabilities would have been linked to hydrodynamic instabilities propagating in the upstream direction, they would have been visible in the space-time cross correlation of axial velocity in Figure 7 as positive slope lines of high amplitudes. It can thus be concluded that those instabilities are linked to acoustic waves propagating in the upstream direction. A second zone is visible for 12h < z < 15h, with a negative slope, i.e. with downstream propagating instabilities. This is the result of the screech feedback mechanism establishing mainly between the nozzle lips and the end of the tenth shock cell, at z = 12h. Finally, for the case JetL0, only one pattern is visible with an inclined line of strong cross correlation in red. The slope is negative, indicating a downstream propagating instability. This slope simply corresponds the the convection velocity. Indeed, as JetL0 does not show a dominant flapping motion, no clear organization can be seen.

The results for the upper shear-layer are given in Figure 11. Qualitatively, the results are similar to the one obtained for the lower shear-layer in Figure 10. However, for JetL1 and JetL2, althought the pattern is the same, higher levels of cross correlation are found in the lower shear-layer in Figure 10 compared to the upper one in Figure 11. This result suggests that the amplification of the feedback mechanism creating the screech noise is due to a stronger feedback establishing in the lower jet shear-layer. In this paper, compressible LES of a rectangular over-expanded supersonic jet interacting with a parallel wall plate are performed. The jet exits from a converging diverging rectangular nozzle of aspect ratio 2 and of design Mach number 1.5. Four simulations with four different distances between the lower inner lip of the rectangular jet and the plate ranging from 0 to 3 equivalent diameters downstream from the inner lip in the minor axis plane are carried out. The geometry of the nozzle, the positions of the plate, and the exit conditions are chosen in order to match those in an experimental study conducted at the University of Cincinnati. Aerodynamic fields are first presented, contrasting the LES mean fields with experimental PIV results. A good overall agreement is reached. The overall sound pressure levels are then plotted along the minor and major axis. It is seen that the screech feedback mechanism is almost suppressed when the plate is located at the inner nozzle lip position. Moreover, its intensity is amplified for some distances between the jet and the plate and weakened for some others. These observations are in good agreement with the ones made by the experimental team. A study of the jets shear-layers is then proposed first by looking at two points space-time cross correlations of the axial velocity. The convection of the turbulent flow structures in the jet shear-layers is thus studied. Then, two points space-time cross correlations of the pressure along the jets shear-layers are proposed and an amplification of the aeroacoustic feedback mechanism leading to screech noise is observed in the lower jet shear-layers for two cases. It is also observed that the screech feedback mechanism establishes mainly between the nozzle lips and the end of the tenth shock cell, at z = 12h. The acoustic loading on the plate is finally studied. As pointed out in a previous study, the flapping motion of the jet at the screech frequency seems to yield to an asymmetric organization of the Mach wave radiation also at the screech frequency. The organized Mach waves impinge on the plate and propagate back towards the jet, exciting the shear-layer at the screech frequency. This feedback amplifies the screech mechanism in the lower jet shear-layer. However, this amplification happens only for some nozzle-to-plate distances. Indeed, the screech mechanism leads to the formation of a standing wave pattern in terms of pressure loading at the screech frequency on the plate. There are regions with high amplitude, meaning the acoustic loading is organized mainly at the screech frequency, and regions with low amplitude, which means the acoustic loading is not organized mainly at the screech frequency. The amplification thus depends on the location of the standing wave compared to the overall acoustic loading on the plate. If a region of high amplitude of the standing wave pattern coincides with the region of maximal acoustic loading, there is an amplification of the screech mechanism.

Figure 1 .

 1 Figure 1. Isosurfaces of density for the value 0.7 kg.m -3 colored by the local Mach number for JetL2.The nozzle and the plate are in darkgrey.

Figure 3 .

 3 Figure 3. Representation of the Mach number in the jet and of the fluctuating pressure around for (a,b) JetL0, (c,d) JetL1, (e,f ) JetL2 and (g,h) JetL3 in (a,c,e,g) the minor axis plane and (b,d,f,h) the major axis plane. The nozzle and the plate are in black.

Figure 4 .

 4 Figure 4. Mean axial velocity fields for (a,e) JetL0, (b,f ) JetL1, (c,g) JetL2 and (d,h) JetL3 in (a,b,c,d) the minor axis plane and (e,f,g,h) the major axis plane. The experimental PIV results are added in black rectangles. The nozzle and the plate are in black.

Figure 5 .

 5 Figure 5. Overall Sound Pressure Levels for (a,e) JetL0, (b,f ) JetL1 (c,g) JetL2 and (d,h) JetL3 in (a,b,c,d) the minor axis plane and (e,f,g,h) the major axis plane. The nozzle and the plate are in black.

4

 4

Figure 6 .

 6 Figure 6. Acoustic spectra in the vicinity of the nozzle at (x, y, z) = (0, h, 0) for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The acoustic spectrum of the corresponding free jet is added in lightgrey for comparison purposes.

Figure 7 .

 7 Figure 7. Two points space-time cross correlations of the axial velocity along the lower shear-layer in the minor axis plane as a function of the axial coordinate of the second point for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The reference point is located at the z = 7.5h and the second point extends from z = 0 to z = 15h. The color scale ranges from -1 to 1.

Figure 8 .

 8 Figure 8. Two points space-time cross correlations of the axial velocity along the upper shear-layer in the minor axis plane as a function of the axial coordinate of the second point for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The reference point is located at the z = 7.5h and the second point extends from z = 0 to z = 15h. The color scale ranges from -1 to 1.

Figure 9 .

 9 Figure 9. Convection velocity of the turbulent structures along the upper ( ) and lower ( ) shearlayer in the minor axis plane as a function of the axial coordinate for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The convection velocity of the turbulent structures of the corresponding free jet is added in lightgrey for comparison purposes.

Figure 10 .

 10 Figure 10. Two points space-time cross correlations of the pressure along the lower shear-layer in the minor axis plane as a function of the axial coordinate of the second point for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The reference point is located at the z = 7.5h and the second point extends from z = 0 to z = 15h. The color scale ranges from -1 to 1.

Figure 11 .

 11 Figure 11. Two points space-time cross correlations of the pressure along the upper shear-layer in the minor axis plane as a function of the axial coordinate of the second point for (a) JetL0, (b) JetL1, (c) JetL2 and (d) JetL3. The reference point is located at the z = 7.5h and the second point extends from z = 0 to z = 15h. The color scale ranges from -1 to 1.

Table 1 .

 1 Jet parameters: Plate position L, Nozzle Pressure Ratio N P R, Mach number of the ideally expanded equivalent jet Mj = uj /aj , Temperature Ratio T R and exit velocity of the ideally expanded equivalent jet uj . aj is the speed of sound in the ideally expanded equivalent jet.

	-1

Table 2 .

 2 St low dB low St screech dB screech Frequencies and amplitudes for the four present jets and for the corresponding free jet.

	free jet 0.18	145.6	0.315	159.5
	JetL0	0.13	143.8	0.32	142.4
	JetL1	0.165 139.8	0.32	161
	JetL2	0.19	141.3	0.325	163.5
	JetL3	0.19	145.2	0.325	155

Figure 2. Isosurfaces of density for the value 0.7 kg.m -3 colored by the local Mach number for JetL3.The nozzle and the plate are in darkgrey.

Acknowledgments

The computations were performed using HPC resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC center. Florian Baier, PhD at the Department of Aerospace Engineering & Mechanics of the University of Cincinnati is acknowledged for providing the experimental data used in this paper.

VI. Organization on the plate

The fluctuating pressure on the plate is recorded along time at a sampling frequency of St = 2.3. A first view of the data is proposed in Figure 12 with snapshots of the fluctuating pressure on the plate. For the four jets, the acoustic component of the fluctuating pressure is visible with acoustic waves organized in the same way as in the near acoustic fields. Moreover, the hydrodynamic component of the fluctuating pressure can also be seen in the jet region for JetL0 and for JetL1 at z > 10h. This hydrodynamic component is due to the interaction between the jet plume and the plate in these two cases. Two post-treatments are applied to the fluctuating pressure fields on the plate. First, the root mean square value is computed in order to look at the locations of maximum acoustic loading on the plate. Then, a Fourier decomposition is also applied to those fields, permitting to look at the organization of the pressure loading at the screech frequency. Results are plotted in Figure 13 for the four jets at the screech frequency.

For the three jets undergoing a strong screech mechanism, the maxima in the root mean square of the fluctuating pressure are located on the jet axis, as expected, at z = 2.78h for JetL1, z = 5.3h for JetL2 and z = 6.82 for JetL3. In the amplitude fields, a standing wave pattern can be seen for those three latter jets. Those waves are hydrodynamic-acoustic standing waves created by the screech mechanism. Such structures were previously observed by Panda et al. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] for screeching supersonic jets and by Gojon et al. [START_REF] Gojon | Investigation of tone generation in ideally expanded supersonic planar impinging jets using large-eddy simulation[END_REF] for ideally expanded planar impinging jets. In this standing wave pattern, there are regions with high amplitude, meaning the acoustic loading is organized mainly at the screech frequency, and regions with low amplitude, which means the acoustic loading is not organized mainly at the screech frequency. It can be seen that the localization of the maxima in the root mean square fields coincide with a region of high amplitude for JetL1 and JetL2 but not for JetL3. This suggests that for JetL1 and JetL2, where the acoustic loading of the jet is maximal on the plate, the acoustic waves are organized mainly at the screech frequency, leading to reflective acoustic waves also organized along the screech frequency. These acoustic waves will then excite the jet shear-layer at the screech frequency, leading to an amplification of the screech mechanism in the lower jet shear-layer. This result explains the stronger feedback mechanism establishing in the lower jet shear-layer compared to in the upper jet shear-layer pointed out in Figures 10 and11 for JetL1 and JetL2.

In order to have further insights of the content of the acoustic loading at the points on the plate where the root mean square value of the fluctuating pressure is maximum, the pressure spectra at those points are represented in Figure 14 for those three cases. In Figure 14, for JetL1, a strong peak can be seen at St = 0.32 at an amplitude of 158 dB/St. For JetL2, a peak can be seen at St = 0.325 at an amplitude of 156 dB/St. The results indicate that for the two jets the acoustic loadings at the specified positions are mainly organized along the screech frequency. However, for JetL3, the screech mechanism is still dominant in the near acoustic field, as observed in Figure 6, but cannot be seen in Figure 14 at the position where the acoustic loading on the plate is maximal. In conclusion, it is observed that if the acoustic waves are organized at the screech frequency at the position of the maximal loading on the plate, an amplification of the screech mechanism is observed. It is also worth noting that this amplification mechanism is in line with the observations made on the space-time cross correlation of the fluctuating pressure in the jet shear-layer in Figures 10 and11 where it has been observed that the screech mechanism possesses a higher amplitude in the lower jet shear-layer than in the upper jet shear-layer for JetL1 and JetL2.