Samuel Bishop

Wilberforce

Henry John

Newman's

Samuel Wilberforce

John Henry Ker

Newman

Jérôme Grosclaude

"AN AMAZING WANT OF CHRISTIANITY"

Keywords: church of england, theology, united kingdom, john henry newman, anglicanism

This paper will examine the relationship between Samuel Wilberforce and John Henry Newman. The two priests had a common cause in their wish to see the Church of England rediscover its Catholic identity -which led them to work alongside one another at the beginning of the Oxford Movement -but quickly drifted apart because of their strong divergences on the nature of the Church and the place of Tradition, as well as Samuel Wilberforce's strong hostility to Rome. The paper also examines the place of Samuel Wilberforce's young brother in this relationship.

That the two men should not have known each other more may seem surprising since they began as virtual allies in the revival of Catholic ideas in the Church of England; and had in common the love and respect of Robert Isaac Wilberforce (1802-1857), Samuel's beloved older brother and a disciple of Newman.

However, I will show that, close as they were in their high view of the ministry (and particularly of the episcopate), Samuel Wilberforce and John Henry Newman parted ways over their conceptions of the Church, and of Tradition. We will first see how Samuel Wilberforce and John Henry Newman associated with the same people at Oxford in the 1820s and the 1830s (I); from the middle of the 1830s, Samuel Wilberforce's emphasis on the Protestant character of the Church of England, separated him more and more from John Henry Newman and his disciples (II); finally, we will look to the fundamental doctrinal differences which existed between John Henry Newman and (III).

I. MISSED OPPORTUNITIES? a) Similar influences

It should first be noted that the two men were only a few years in Oxford (at Oriel College) at the same time. Samuel Wilberforce arrived at Oriel as an undergraduate in early 1823 5 ; he graduated BA in the Michaelmas Term of 1826; the following year he left England to take an extended foreign tour of several months until November 1827 6 , he seems not to have come back to Oxford before December 1828 when he was ordained a deacon. Three months later he was in his first curacy, at Checkendon, 20 miles south from Oxford.

He did come back to Oxford in the course of the next sixteen years, for specific 4 Life, vol. II, 261-262 5 Life, vol. I, 26. 6 Ibid., vol. I, 36. By contrast, John Henry Newman, who was Samuel Wilberforce's senior by more than five years 8 , entered Trinity College in 1817 and was elected to an Oriel Fellowship in 1822. He was to remain in Oxford until 1843 when he resigned his position at Saint Mary's to live a quasi-monastic life at Littlemore.

Two years later, Samuel Wilberforce was appointed bishop of Oxford, where he was to remain for the next 24 years. John Henry Newman himself did not come back to Oxford before 1876, and by that time, Samuel Wilberforce has been dead for two years.

During these few years, the two men do not seem to have met in any significant way: Samuel Wilberforce testified to this in 1834 when, then a young clergyman on the Isle of Wight, he wrote to Newman: "I wish that … you would … allow me to repair one of my great Oxford faults -that of neglecting the endeavor to obtain a more intimate acquaintance with you 9 ". He however invited John Henry Newman to "a few days' relaxation" on the Isle of Wight in July 1836 10 while his brother Robert was present (J. H. Newman declined 11 ).

With more precision, John Henry Newman was to write in 1876 to Canon Arthur R. Ashwell who had written to ask him for any correspondence he 7 The Convocation is the legislative body of the University of Oxford, composed of "all the former student members of the University who have been admitted to a degree (other than an honorary degree) of the University" as well as current and retired office-holders of the University (cf. https://www.ox.ac.uk/about/organisation/history/oxford-glossary?wssl=1 (accessed 31.5.2019). 8 John Henry Newman was exactly 5 years, 6 months and two weeks older than Bishop Wilberforce. 9 David Newsome, The Parting of Friends, 97. and sola scriptura -all elements often associated with Evangelicalism -, as well as adhering to a high view of the Church, of episcopacy and of the sacraments. In 1840, he noted in a lengthy letter to a fellow priest who had accused his latest collection of children's stories, The Rocky Island, of reflecting Tractarian beliefs, that his opinions has been moulded "in a far different school": "they are those of my beloved father, as I could prove, were it needful, from many written records of his judgment as to the tenor of my ministry, of which, during his late years, he was a most kind, but a close observer 15 ". John Henry Newman also came from an Evangelical background 16 , although he later identified his parents' Evangelicalism as merely "'the national religion of England', 'Bible religion', which 'consists not in rites or creeds, but mainly in having the Bible read in Church, in the family, and in private 17 '"; at fifteen, John Henry Newman had a conversion, in which he became acutely aware of the existence of God. He wrote that, before that, his general frame of mind had been a wish "to be virtuous, but not religious. There was something in the latter idea I did not like. Nor did I see the meaning of loving God 18 ". However, from the following year, he "fell under the influence of a definite creed, and received into [his] intellect impressions of dogma, which, through God's mercy, have never been effaced or obscured 19 ".

A lack of personal alchemy

Despite these common elements (the love of Robert Wilberforce and Evangelical nurturing) Samuel Wilberforce never enjoyed nor seems to have actively sought being among the circle of disciples of John Henry Newman. It was not at the start because of any prejudice against John Henry Newman, but 15 Life, vol. I, 217. 16 Ian Ker, John Henry Newman, 3-5. 17 Ibid., 3. 18 Ibid., 4. 19 Ibid., 4. 6 probably because of a lack of personal alchemy, and also because of the conventions of the time which did not easily provide for circumstances in which an undergraduate would have been able to establish personal relationships with a Fellow and a College tutor. As a matter of fact, John Henry Newman dismissed Samuel Wilberforce in a private letter written in November 1838: "Samuel W[ilberforce] is so far from anything higher than a dish of skimmed milk that we can hope nothing from him 20 ." In another letter (written in September 1841), he dismisses him as the man "whom Froude and I have stigmatized as a humbug for many years 21 ".

Nonetheless, the two men could have been expected to be on the same side in the religious battles of the 1830s. On the general principles of the fight, they agreed on the importance of episcopacy. In his first published sermon, preached in 1833, Samuel Wilberforce urges priests to "prize at a higher rate that unbroken succession whereby those who ordained us are joined into Christ's own Apostles", as well as to avoid the "danger of quitting the high vantage ground of Apostolical authority to fight the battle out upon the doubtful level of Erastian principles 22 ."

The very same year, in September, the first of the Tracts for the Times, written by John Henry Newman, was published, emphasizing as it did the episcopal ministry in its conclusion:

A notion has gone abroad, that they can take away your power. They think they have given and can take it away. (…) Enlighten them in this matter. Exalt our Holy Fathers, the Bishops, as the Representatives of the Apostles, and the Angels of the Churches; and magnify your office, as being ordained by them to take part in their Ministry 23 . 20 -am convinced by it, if not of the duty, yet certainly of the expediency of conforming to the Rules of the Church on this point. I think it likely to be especially useful to me (…) 25 ". He was also (probably) distributing them as evidenced by a letter of June 1836 where he asks Newman: "will there be a fresh supply of Tracts on July 1? I wish to know that I may order" 26 .

However, when one turns to the practical aspects of the fight, one sees distance getting more and more obvious. The first skirmish was the Hampden affair, which started in late January 1836, when King William IV, acting on the recommendation of the Prime Minister, the 2 nd Viscount Melbourne, appointed Renn Dickson Hampden (1793-1868) to the vacant Regius Professorship of Divinity at Oxford University. Dr Hampden's orthodoxy was questioned, mainly on the basis of his Bampton Lecture of 1832, and of his Observations on Religious Dissent (1834). For Owen Chadwick, Hampden's writing was "dull" and his statements "often vague and obscure" 27 . Certainly, Samuel Wilberforce agreed with this, who observed jocularly in a 12/2/1836 letter to his brother Robert: "could we not pass a vote that Hampden should always preach in Hebrew? 28 ". John Henry Newman, for his part, was scandalised at Hampden's alleged latitudinarianism which led the prospective Professor, notably, to draw a distinction between the original text of Scriptures and the various dogmatic formulas introduced (or should one say "forced"?) into it over time 29 . John Henry Newman observed in a private letter written at the time: "there is no doctrine, however sacred, which he does not scoff at -and in his Moral 24 Probably Tract LXVI (13/4/1835). 25 Philosophy he adopts the lowest and most grovelling utilitarianism as the basis of Morals -he considers it is a sacred duty to live to this world -and that religion by itself injuriously absorbs the mind 30 ." Seventy-three residential fellows and tutors on the one hand, and nine heads of houses on the other signed petitions of protest. After many articles, petitions and angry letters, Hampden's appointment was duly gazetted on 20 February, twelve days after it had unofficially become public knowledge 31 .

Convocation then moved to limit as much as possible Hampden's prerogatives: a motion of no confidence in Hampden passed by the Board of Heads was passed on 2 May (by 474 to 94) 32 . Samuel Wilberforce, who had pronounced Hampden's rumored appointment in the same 12 February letter "disgusting 33 ", convened some of his colleagues in his rural deanery to draw up some sort of protest (however nothing seems to have come out of it) 34 . He was evidently present at Convocation in March to vote against Hampden and seized the opportunity to have some very long conversations with Newman upon several of the most mysterious parts of the Christian Revelation, the Trinity, &c., as well as upon some of the greatest practical difficulties to faith arising from the present torn state of Christendom; and it was really most sublime as an exhibition of human intellect when in parts of our discussions Newman kindled, and poured forth a sort of magisterial announcement in which Scripture, Christian antiquity deeply studied and thoroughly imbibed, humility, veneration, love of truth, and the highest glow of poetical feelings, all impressed their own pictures upon his conversation 35 .

Unsurprisingly, Samuel Wilberforce pronounced himself "delighted" at the meeting. John Henry Newman and Samuel Wilberforce found themselves 30 However, as soon as 1838, the two men parted way, quite radically so.

So the two men, whose upbringing and intellectual itineraries and preoccupations should have brought them together however quickly drifted apart, mainly because of a lack of personal alchemy, but maybe more than that, because of a fundamental difference on doctrinal questions.

II. SAMUEL WILBERFORCE: THE TRACTARIAN THAT WASN'T a) Doctrinal proximity

As many High Churchmen, Samuel Wilberforce was regularly -almost routinely -accused of being a Tractarian, or a Papist, or both, in disguise. You do not, I hope, [think] that I belong to the school of the "Tracts for the Times". I admire most highly the talents of some of those men: I revere far more their high and self-denying holiness and singleness of purpose: but I cannot agree with them in all their leading views of doctrine (e. g. Pusey's, as far as I understand it, view of Sin after Baptism), and I often find in practical matters that I differ from them, on points, and in ways, (…) in which, as it seems to me, they are for enforcing an ancient practice at the expense of a still more ancient principle 40 .

The following month, in a belated reaction to Pusey's Tracts on Baptism, Samuel Wilberforce preached on 18 February 1838 his first sermon as select preacher before the University 41 . The sermon was entitled "The Penal consequences of Sin" and was clearly intent on balancing the harsh view expressed in the sermon I mentioned previously and that he had preached barely a year before. In the published version of 1839, where they represent the first two sermons, Samuel Wilberforce took care to make clear that both texts "should be read together: for some expressions of the first, taken by itself, might seem to favour that view which the second is specifically intended to counteract 42 ". The second sermon, indeed, took as its text the conclusion of the parable of the prodigal son in which, he said, "all still speaks it the description great may be one man's sin, the Heavenly Father's love would still be greater 44 .

And even to the first acts of a sincere penitence, surely there are here promised some gracious marks of acceptance, as what shall be given to the returning sinner. Baptised, indeed, he cannot be afresh: but does he lose by that? No, truly; (…).

[A]nd now, if he comes in sincerity and faith, the seal is still sure, and is for him ; his baptism is on him, fresh as when its waters glistened upon his infant brow ; he is received into his Father's house; and there the words of gracious promise, the blessed seals of holy eucharists, and the fresh-springing fountain of the Saviour's blood, these are sure and for him; and they are meant to carry to his soul the same certain consolation which the holy waters of baptism would be the outward means of bringing, if he came as a catechumen, instead of coming as a penitent 45 .

Such words were widely seen as rebuking Tract LXVII's stringent view of sin after baptism, in which Pusey notably wrote (my emphasis):

For our modern system, founded, as it is, on the virtual rejection of Baptism as a Sacrament, confounds the distinction of grievous sin before and after Baptism, and applies to repentance, after falling from Baptismal grace, all the promises which, in Scripture, are pledged, not as the fruit of repentance simply, but as God's free gift in

Baptism. Yet our reformers thought differently; for had their theology been like ours, there had been no occasion for an article on "Sin after Baptism" (Art. XVI), or for denying that "every such sin is sin against the Holy Ghost, and unpardonable". It had been a matter of course. The possibility or efficacy of such repentance I have not denied; God forbid: but that such repentance is likely, especially after a relapse, or that men, who have fallen, can be as assured of the adequacy of their This episode found clearly differentiating himself from the Tractarians: he had at first intended to vote for Williams, on the basis of his poetry credentials and of "old friendship 54 ". In a letter to a friend of his written in November 1841 he curiously states that he had at least intended not to vote against him 55 . However, in his view, Pusey's letter had "altered the circumstances of the case": "I cannot hide from myself that now it must be, Newman's view of Justification; the language of Tract 90, the "British Critic" &c., as to Rome; the craving after unity through some visible centre; the saying that old Rome was that centre (…); the fearful doctrine of Sin after baptism, the whole tone about the Reformers, &c. &c.-all this has pained me and grieved me so entirely, that I have felt daily obliged more and more, from love of the truth as I saw it, from love to our Church, whose principles and very life I believe this teaching threatens, (…) to take on all occasions a position of more direct opposition to the School than I had of old thought necessary 61 . Apart from the possible question of purely personal alchemy (or lack of it) between John Henry Newman and , I previously noted that the differences between the two men was due to doctrinal differences. I think we can trace the root of these differences of viewpoints to the two men's visions of the Church.

III. WHAT IS THE CHURCH?

a) The Episcopate The first tract, authored by John Henry Newman, explicitly addressed to priests of the Church of England, adopted an energetic tone to remind them of what the apostolic succession was, as well as the fact that the doctrine was explicitly professed by the Church of England:

I
[E]very one of us believes this. I know that some will at first deny they do; still they do believe it. Only, it is not sufficiently practically impressed on their minds. They do believe it; for it is the doctrine of the Ordination Service, which they have recognised as truth in the most solemn season of their lives 64 .

John Henry Newman, for one, had an exalted view of the Episcopal office. In his pamphlet, The Restoration of Suffragan Bishops (1835), he argued that the clergy were not "mere instruments and adjuncts of the State 65 " but "those who are by office guides of conduct, arbiters in moral questions, patterns of holiness and wisdom, and not the mere executive (sic) of a system which is ordered by prescribed rules, and can go on without them 66 ". He adds that the bishop was really "the centre and emblem of Christian unity, the bond of many minds, and the memento of Him that is unseen 67 ".

Later on, John Henry Newman developed the "Branch theory":

The Catholic Church in all lands had been one from the first for many centuries; then, various portions had followed their own way to the injury, but not to the destruction, whether of truth or of vanity. These portions or branches were mainly three : -Greek, Of course, by the time wrote this, John Henry Newman had joined the Roman Catholic Church and an actual gulf now existed between them.

In fact, on both counts, and under the harshness of the denunciation, we But wasn't there something else at play when it comes to his actual relationship with John Henry Newman? Wasn't there something like jealousy in seeing his dear brother getting so close to John Henry Newman? At any rate, the year 1851 saw an actual struggle by proxy between Newman and S.

Wilberforce, whose object was the soul of Robert Wilberforce. On 12/9/1854, writes to his brother (replying to the letter in which he had announced his conversion): "I see that originally J. Newman obtained a great power over your mind; that since, through your great humility, Manning, by his great subtlety of intellect, and Henry by his unceasing repetition of argument, have overmastered your own far superior understanding 89 ".

Was there, at the bottom of all of this, bitterness at not having fully understood Robert, whom he called in a letter written to him some time before his conversion: "my brother and friend, --my friend, guide and aid since boyhood 90 "? It is difficult to judge, of course, but there may well have been, at the bottom, of this strained relationship between John Henry Newman and , the unhappy love of a disappointed brother.

  me for the rest of my life against the cursed abominations of the Papacy 4 .

  preaching sermons at Saint Mary's or to cast votes as a member of Convocation 7 . Significantly, he was not living in Oxford during the crucial years of 1833-1841 when the Tracts for the Times were published. He was dean of Westminster Abbey for six months before being appointed bishop of Oxford in November 1845.

  with the late Bishop of Oxford (of whom he was writing the biography):As to any correspondence I have had in former years with Bishop Wilberforce, it never was much -but I will look out for you any letters of his I can find. He is the one out of the four brothers whom I never was intimate with. I knew him from the time he came up to Oxford in October (say) 1823, and we had friendly relations with each other from my affection for his brother Henry -but such kind interest as men so different in all respects might have in each other received a rude shock never recovered from on his preaching in the University Pulpit against Pusey's views of Baptism about 1836-38 12 .However, Samuel Wilberforce, just like his brother Robert 13 , was quickly drawn towards the leaders of what was soon to be called Puseyism: as many of their contemporaries, the Wilberforce brothers had been seduced by John Keble's Christian Year (1827). Samuel Wilberforce remained all his life in awe of John Keble (1792-1866) whose word always had enormous weight with him -even when he had become a bishop. Samuel Wilberforce also deeply respected Pusey's learning and piety. In several letters, while lamenting what he saw as the Tractarian leaders' want of discretion, always took pains to observe that he respected their devotion and practice of Christianity 14 .The Wilberforces came from an Evangelical household. It may be surprising that, out of William Wilberforce's four sons -of which three became clergymen -, three converted to Roman Catholicism and the only one who did not nonetheless side firmly with the High Church and was time and again accused of being a Papist in disguise. Until the end, Samuel continued to cherish the memory of his father, and did not see any contradiction in preaching12 John Henry Newman to Arthur Rawson Ashwell, 30 March 1876, LD xxviii, 46. 13 Samuel Wilberforce's younger brother, Robert Wilberforce, was a Fellow of Oriel College, and had been a tutor along with Newman and Hurrell Froude from 1826 to 1831. Originally closer to Pusey, Robert Wilberforce was drawn to Newman because of their common dedication to giving their Oriel tutorships a pastoral aspect (David Newsome, The Parting of Friends, 92 ff.). 14 Life, vol. I, 217.

  s Evangelical emphasis on private devotion and self-examination

  fight, even though it was not only High Churchmen, but also Evangelicals 36 , who opposed Hampden's appointment. The Archbishop of Canterbury himself, William Howley, noted publicly in the House of Lords on 21/12/1837 that Hampden's appointment was the only Crown appointment he had ever objected to 37 ". That was really the high water mark of Samuel Wilberforce's opinion of John Henry Newman. Of this honeymoon between the two men, Samuel Wilberforce's Lent sermon of 1837 before the University bears testimony. Taking as its text Second Corinthians VI, 1 38 , exposed a view of Justification and Sanctification deeply tributary to John Henry Newman's Lectures on Justification published this very same year. Emphasizing the gifts of Baptism (in a manner perfectly consistent with the XXXIX Articles), he pointed the fearful dangers of sin after baptism by observing: "you may reclaim the sinner from open vice; you cannot renew him to holiness 39 "

  of a fallen son of the Most High 43 ". He argued that, however

  doubt that the Memorial was indeed a party project (even though his main promoter, C. P. Golightly, was a traditional High Churchman), and s remark that the Tractarians should have contributed to a project implicitly intended at embarrassing them seems, as best, naive. S. Wilberforce's main argument in favor of such a move, however, was the need of showing unity in the defence of the Church -always a very potent argument in view. Three years later (in the Autumn 1841), John Keble was to retire from the University's Chair of Poetry, and the vote by Convocation to choose his successor was scheduled for January 1842. Isaac Williams (1802-1865) was widely considered as being due to succeed him; however, his long association with the Tractarian leaders could hardly be passed over in the deeply polarized Oxford of the time: he had published many times in the British Magazine and the British Critic (the latter an obvious vehicle for Tractarian ideas), had been a curate to Newman and had contributed three Tracts between 1837 and 1840. Thus, an opponent was quickly found in the person of the Evangelical James Garbett (1802-1869); although he had published no poetry, contrary to Williams. Garbett's candidacy was helped by what some saw as the necessity of checking the Tractarians' progress. A letter by Pusey injudiciously attacking Garbett and explicitly supporting Williams turned the election into a referendum on Tractarianism in which the poetry credentials of the two candidates were becoming irrelevant.

  was thus affirming himself as a representative of a new High Church, distinct from the traditional High Church of, say, Bishop Henry Phillpotts of Exeter (1778-1869) (with its strong association with the Tory party) as it valued involving oneself in the work of the Commonwealth, while valuing the ordained ministry, and particularly the episcopate, and at the same time keeping jealously one's distance with Rome. was also quietly but staunchly attached to the Protestant identity of the Church of England.

  will not enter here in a detailed dissertation on what the Tractarians stood for or against; others have done it much more brilliantly than I could. Let me however remind the reader, if it were necessary, that the Tractarians put much emphasis on the Episcopal ministry and the apostolic succession. Even before the first Tract was published, William Palmer of Worcester College wrote: The bishops who rule the churches of these realms were validly ordained by others, who by means of an unbroken spiritual descent of ordinations derived their mission from the apostles, and from our Lord. (…) Our ordinations descend in an unbroken line from Peter and Paul, the apostles of the circumcision of the Gentiles. (…) The true and orthodox bishops and pastors teach those doctrines which the catholic church has taught in all ages from the beginning 62 . 62 William Palmer, Origines Liturgicae, or Antiquities of the English Ritual (Oxford: Oxford University Press, 1832), vol. II, 249 & 253.

  Palmer would make the exact same claim in the only tract he authored, Tract 15 (13/12/1833), entitled On the Apostolical Succession in the English Church 63 .

  have the impression that lamented that the Tractarians should go too far in what he believed to be the right direction. This is how I, for one, reads 's mixed description of the Oxford movement in July 1869, when reviewing Sir John Taylor Coleridge's biography of John Keble : "a great religious movement, which is still more than any other affecting for good, or for evil, or for both, the present and future tone of the Church of England 87 ".CONCLUSIONNineteenth-and twentieth-century biographers of Samuel Wilberforce have tended to interpret life as a struggle for true High Church principles against the Romanizing tendencies of the Tractarians and their heirs. This is basically the thesis of George William Daniell's Bishop Wilberforce (1891) who presents the Bishop as "the remodeller of the episcopate", a man aware85 Standish Meacham, Lord Bishop, 90.86 Ibid., 90-91. 87 "Keble's Biography", 231.

  is kept afar off by the development, and not by the terror-stricken suppression, of the true Catholic idea 88 ". The third biography of the Bishop of Oxford, John Charlton Hardwick's Lawn Sleeves: A Short Life of Samuel Wilberforce (1933), is full of bitter attacks against the Oxford Movement which, in 1933, was then exactly 100-year old. And indeed S. Wilberforce's bitterness against the Tractarians was clearly influenced by his dismay at seeing honorable men promoting muchneeded forgotten truths about the Church while (in his view) mixing them up with Romanizing tendencies.

  Life, vol. I, 83. 26 Quoted in Standish Meacham, Lord Bishop, 67. 27 Owen Chadwick The Victorian Church, Part I: 1829-1859 (London: SCM Press, 1987), 116.

28

Life, vol. I, 93.

29

Owen Chadwick The Victorian Church, Part I, 116.

  John Henry Newman to Simeon Lloyd Pope, 3 March 1836, LD v, 251. 31 Ibid. 118. 32 Life., vol. I, 93. 33 Ibid., vol. I, 93.

34

Ibid., vol. I, 93.

35

Samuel Wilberforce to Louisa Noel, 1 April 1836, ibid., vol. I, 93.

as they might have been of God's free grace in Baptism, daily experience, as well as the probable meaning of Scripture, forbid us to hope 46

  . John Henry Newman very rightly considered Samuel Wilberforce's sermon as directly aimed at Pusey's (and the Tractarians') conception of Justification. Five months later, on July 18 th , John Henry Newman wrote to Samuel Wilberforce to decline any further contribution from him to the British Critic 47 : To say frankly what I feel -I am not confident enough in your general approval of the body of opinions which Pusey and myself hold, to consider it advisable that we should cooperate very closely. (…) [A]nd though I feel we ought to bear differences of opinion in matters of detail, and work together in spite of them, it does not seem to me possible at once to oppose and to cooperate; and the less intentional your opposition to Pusey on a late occasion, the more impracticable does co-operation appear 48 .

	repentance,
	F
	o r
	P
	e e r
	R e v
	i e w
	43 Ibid., 38.
	44 Ibid., 39.
	45 Ibid., 40.
	12

Strangely enough, Samuel Wilberforce appeared very disappointed by the letter, and saw this refusal as a manifestation of petty party spirit 49 .

b)

The Roads that don't lead to Rome Pusey

	time at Oxford, and it seems that Froude was the first person to whom he
	mentioned his desire to seek orders 50 .S. Wilberforce was all the more
	astounded by the book when it came out. His letters and diary come back again
	and again on the Remains which clearly startled him and deeply repelled him:
	in his diary, at the date of 25 March 1838, he writes thus: "Evening, read a little
	of Froude's 'Journals'. They are most instructive to me. Will exceedingly
	discredit Church principles, and show an amazing want of Christianity, so far.
	They are Henry Martin unchristianized 51 ."
	The publication of Froude's mostly private papers, in which he made
	F
	o r
	P
	e e r
	R e v
	i e w
	S. Wilberforce appears to have been seeking Froude's company during his
	50 Life, vol. I, 35-36.
	51 Ibid., vol. I, 120. This strange reference to the missionary priest Henry Martyn (1781-1812),
	who had died of fever in present-day Northern Turkey seems to indicate the vanity of human
	46 Tract for the Times n° 67 : Scriptural views of Holy Baptism (24/8/1835), § xiii suffering offered in vain and for no purpose.
	47 He had previously written some reviews for them (Standish Meacham, Lord Bishop, 66).
	13

's tract on Baptism appears to have been a decisive factor in S. Wilberforce's deserting the Tractarian cause. However, another important element appears to have been the publication, in January 1838, of Hurrell Froude's Remains, edited by J. H. Newman and J. Keble. S. Wilberforce had known Froude well, even though the latter was his senior by two and a half years: they had been undergraduates at Oriel together for a year in 1823-1824.

48

Life, vol. I, 125-126.

49

Standish Meacham, Lord Bishop, 67. remarks such as "The Reformation was a limb badly set-it must be broken again in order to be righted 52 " or "I believe I have a want of reverence, else I should not have got to hate them [the Reformers] so soon as I did 53 ", came as a shock to S. Wilberforce, and seems to have acted as an eye-opener for him.

We can probably date from this moment S. Wilberforce's cutting himself off from the Tractarians, in whom he seems to have discerned a dangerous Rome-ward trend.

Starting from here, S. Wilberforce's break with the Tractarians further materialized itself in the debate surrounding the building, in the Winter of 1838-9, of Oxford's Martyr's Memorial The endeavor had been specifically contrived to embarrass the Tractarians: if they contributed, they would be seen as turning their backs on their beliefs; if they refused (as they ultimately did), they could be denounced as enemies of the Reformation and Papists in disguise. S. Wilberforce pronounced himself in favor of contributing in a letter to his brother Henry dated 29 November 1838.

52

Richard Hurrell Froude, Remains of the Late Reverend Richard Hurrell Froude, M.A., Fellow of Oriel College, Oxford (London: J. G. & F. Rivington, 1838), vol. II, 433. 53 Ibid., vol. II, 435.

  , simply expressing publicly, aye or no, one's approbation of, or dissent from, the most peculiar features of the teaching of the Tract writers. With them, as you well know, I have never agreed 56 ". The following month, he declined 57 to join the endeavor of 257 members of Convocation (including his friend Gladstone, Bishop Phillpotts of Exeter and Bishop Bagot of Oxford) 58 calling on both candidates to desist, as unjust to Garbett (whom he assessed as being most likely to win the election anyway). As a matter of fact, on 20/1/1842, an estimate of pledged votes was made which indicated that Garbett would indeed be elected (921-621), whereupon Williams retired from the contest and the former was elected unopposed one week later 59 . This debate led to strongly condemn the Tractarians in the various letters which he wrote

	whatever one means
	F
	o r
	P
	e e r
	R e v
	i e w
	15

54

Life, vol. I, 205.

55

Ibid., vol. I, 205. at the time. Thus in his already quoted letter: "their views on many points (specially the Tract on Reserve) have appeared to me so dangerous, that, at all costs, I felt I must bear my feeble testimony against them in my Oxford sermons &c. &c. of late, also, they have seemed to me to advance at immense speed 60 ". then went on to list his grievances against the Tractarians:

authority as to teaching with which they invest the early Fathers

  Wilberforce grew to believe that without power to appoint missionary bishops of its own, the Church would be bound in an unseemlyand indeed an unholy -way to the State 70 ". In 1853, he piloted through the House of Lords a (Church) Missionary Bishops Bill ostensibly drafted by the Archbishop of Canterbury John Bird Sumner, but in reality prepared by himself71 , which would have allowed the Church to launch missions to the heathens with English bishops consecrated in England but without the royal mandate. The Bill was adopted by the Upper Chamber but was thrown out in the Commons because of Low Church fears that S. 's hidden agenda was to take over Evangelical missions to force High Church bishops and ideas upon them and their flocks 72 . So shared with the Tractarians an exalted view of the episcopate, which clearly was part of the esse of the Church Catholick. However, clearly differentiated himself from the Tractarians by his vision of what the Church was, and his belief that they transferred to the Church the adoration due to personal union with God inside the greater framework of the visible Church. As mentioned earlier, the apostolic succession he saw as a gift from God bringing, without any doubt, the power of the Holy Spirit to the Christian Community endowed with such a ministry: inside this community, led by a clergy whose origins could be traced back to the Apostles themselves, the believer could and would find God present. For , the believer could not safely be his own guide, nor simply rely on his sole conscience, without running the risk of falling into what called in 1831 "self-idolizing 77 ". Selfidolizing would also be thwarted by the believer's diligent reading of the Bible, which he must acknowledge as the only rule of faith.

	royal mandate for lands outside the British Isles 69 . According to Standish City 73 . John Henry Newman wrote privately that the plan was "to collect a journey and this
	F o r P F o r P F o r communion out of Protestants, Jews, Druses and Monophysites, conforming under the influence of our war-steamers, to counterbalance the Russian influence through Greeks, and the French through Latins 74 ". S. Wilberforce was quietly but certainly welcoming, for Baron Bunsen (whom he saw frequently when at Court) had assured him that the aim of the scheme was to make Prussians accustomed to episcopacy, prior to its introduction in the Lutheran Church: this is how he described it to his brother Robert: "a truly noble plan by which, I trust, on a back current, Episcopacy will flow into Prussia 75 ". F o r Meacham, "In his enthusiasm for the Episcopal institution, S. Wilberforce even P P
	e e r R e v i e w e e r R e v i e w initially supported the Jerusalem Bishopric in October 1841: King Frederick William IV of Prussia and Queen Victoria jointly decided, on the advice of Baron von Bunsen (the Prussian ambassador to the UK from 1840 to 1854 and a close friend to the Prussian King) to establish a Protestant Bishopric in Jerusalem, in order for Protestantism to stand its ground in the Holy City in front of the Latin and Greek Orthodox Patriarchs. This was a way for England (and to a lesser extent Prussia, which was not a colonial power) to gain a foothold in a region, still under Ottoman control, which was at the center of the attention of the European powers. The bishop of Jerusalem was supposed e e r R e v i e w Christ himself. e e r b) The Church instead of Christ I've already mentioned 's debt to his beloved father's Evangelicalism. Indeed, sermons were often intended to reminding his hearers of the main R e v tenets of Evangelical religion: the believer's closeness to God, the importance of personal holiness and a call to private conscience 76 . The believer was to aim at reaching or comforting a personal and loving relationship with his God who had given his only son in sacrifice for the remission of the sinner's faults. So i e w
	to be nominated in turn by each of the two countries, and to adhere to the to follow Christ was not merely a philosophical or intellectual choice but a
	XXXIX Articles and to the Augsburg Confession. The scheme caused alarm thorough personal commitment involving every aspect of one's life.
	among the Tractarians, by appearing to dissolve the sanctity of the Episcopal
	63 [William Palmer], "On the Apostolical Succession in the English Church", Tracts for the office in an official marriage with non-episcopal Lutheranism. It also looked While emphasizing these traditionally Evangelical themes, put forward
	Times, n° 15, 13/12/1833, § 2. as an affront to the Catholic Churches which were already present in the Holy the necessity -also insisted on by High Churchpeople -of putting this spiritual
	64 [John Henry Newman], "Thoughts on the Ministerial Commission", Tracts for the Times,
	n° 1, 9/9/1833, [n. p.]
	65 John Henry Newman, The Restoration of Suffragan Bishops Recommended (London: J. G. 69 Standish Meacham, Lord Bishop, 252. 73 David Newsome, The Parting of Friends, 289.
	& F. Rivington, 1835), 26. 70 Ibid., 252. 74 Ian Ker, John Henry Newman, 235.
	66 Ibid., 15. 71 Ibid., 252.
	67 Ibid., 17. 72 Ibid., 252; and Life, vol. II, 202-203.
	19 20 21

75

Life, vol. I, 198.

76

Standish Meacham, Lord Bishop, 80. However, while holding a high view of the Church, he differed from the Tractarians' view of the Church on two points: Tradition, and what he saw as their deification of the Church. Both points he mentions briefly in a letter to his brother Robert dated 2/2/1842 (my emphasis):

The two leading errors seem to me to be (i) the , which implies the greater purity of celibacy, that fearful lie which has destroyed the sanctity of married life and polluted every female mind in Italy, to say nothing of other consequences; (2) their craving after a visible centre of unity, from a

belief that the Church is to us instead of an absent Christ, instead of a means of His true presence 78 .

  Tract writers fail here: that they speak, and seem to love to speak, ambiguously of the necessity of Tradition, &c. &c. -the tendency of all which (even if they do not mean what is positively erroneous) must be, I think, and is, (1) to lead men to undervalue God's word (…);(2) to lead men to regard the Romish view of Tradition without suspicion and dread 80 .John Henry Newman of course laid down an entirely different vision of the place of Tradition. In Lectures on the Prophetical office of the Church, he wrote: "We agree with the sectaries around us so far as this, (…) to believe that our creed can be proved entirely, and to be willing to prove it solely from the Bible; but we take this ground only in controversy, not in teaching our own people or in our private studies[START_REF] Henry | Lectures on the Prophetical office of the Church, viewed relatively to Romanism and Popular Protestantism[END_REF] ". "We [Anglo-Catholics] rely on Antiquity to strengthen such intimations of doctrine as are but faintly, though really, given in Scripture 82 ", contrary to Protestantism, which "considers it a hardship to have anything clearly and distinctly told it in elucidation of Scripture doctrine, an infringement of its right of doubting, and mistaking and labouring in vain 83 ". Elsewhere in the same book, Newman asserts: "Catholicity, Antiquity, and consent of Fathers, is the proper evidence of the fidelity or Apostolicity of a professed Tradition 84 ." As for the deification of the Church, a reproach already introduced in his , of the Tract system. The Church, I say, separated from the head is substituted for Him! 85 Three years later, in December 1845, observed, about John Henry Newman's Essay on the Development of Christian Doctrine:Newman's book is wonderfully clever and full of Ecclesiastical history, but the most deeply sceptical book I ever read. It is in fact an assertion from beginning to end that all religion is so uncertain except the fact of there being the Church that we can only go to its living authority to convince us there is a God at all 86 .

	F school of the F
	o r o r
	P P
	e e r e e r
	R e v R e v
	i e w i e w
	was not actually hostile to Tradition, even though the main authorities he
	regularly quoted (apart from the Bible) were Hooker and Andrewes 79 whom
	he seems to have seen more as excellent exegetes than as original thinkers. In
	a letter dated 18/5/1842, he writes:
	I believe the Bible, and the Bible only, to be the rule of faith; and I believe, that to
	bring this strongly and sharply out is a matter of the greatest moment. I think the whole
	77 Life, vol. I, 83.
	22 23

78

Life, I, 213.

79

Ibid., II, 69. many

Caesaropapism. He insisted, on the contrary, on the uninterrupted continuity beetween the Apostles' Church and the then Church of England: as I noted, one of his first sermons had included a call to his fellow priests to "prize at a higher rate that unbroken succession Whereby those who ordained us are joined into Christ's own Apostles".

Even before being himself a bishop, saw the episcopate as the fundamental ministry, as the cornerstone of the Church. On the subject of the colonial expansion of the Church, he immediately and constantly supported the creation of Colonial bishoprics. For him, as he argued, in a sermon in May 1837, the Church had the power to appoint and to ordain bishops without a 68 Apo., 65-66.


Stefano Pezzano

Régis Duvigneau

A fully-conservative sliding grid algorithm for compressible flows using an Isogeometric Discontinuous Galerkin scheme

This work aims at developing a high-order, fully conservative, discretization of sliding grids with applications to compressible flows at different regimes (from subsonic to supersonic). The proposed approach combines a discontinuous Galerkin formulation for Navier-Stokes equations with rational representations originating from Isogeometric Analysis, which allows to design a watertight and fully conservative sliding grid algorithm. A verification exercise is first carried out to rigorously establish the convergence rate of the method. Then, the accuracy and robustness are demonstrated for a flow around a pitching ellipse at different regimes. A comparison with a grid deformation technique and a sensitivity study with respect to the location of the sliding interface are also investigated. Finally, the flow around a vertical-axis wind turbine configuration is simulated to show the potentiality of the proposed approach to deal with more complex geometries.

Introduction

A significant number of engineering flow problems involve rotating components. This is, for example, the case of wind turbines, compressors, helicopters, and many others. Often, in such applications, a part of the geometry remains fixed, whereas other parts rotate. A specific strategy is therefore necessary to represent the time evolution of the computational domain. The use of continuous deformation techniques requires dynamic remeshing when confronted with large displacements of the boundaries (1; 2; 3). In this context, efficient algorithms based on Voronoi diagrams have been successfully employed in hydrodynamics simulations (4; 5). In order to avoid remeshing, the simulation of flows involving rotating objects is frequently carried out using sliding grids, in which the computational domains is subdivided into a fixed and a rotating region. As a consequence of the discontinuous mesh movement between the fixed and the moving subdomain, hanging nodes appear along the sliding interface. Moreover, when piecewise linear grids are employed, the computational domain is characterized by gaps and overlaps. Therefore, specific numerical methods are required to treat the issues introduced by the sliding movement (6; 7; 8).

In this context, Discontinuous Galerkin (DG) schemes are appealing because they are natively capable of handling geometric non-conformities. Ferrer et al. [START_REF] Ferrer | A high order Discontinuous Galerkin -Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes[END_REF] proposed the first DG method with sliding grids for incompressible flows. On the other hand, Isogeometric Analysis (IGA) [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[END_REF] is well adapted to rotating grids because circular interfaces can be exactly represented, thanks to the Non-Uniform Rational B-Splines (NURBS) representation, avoiding the emergence of holes and overlaps. In the framework of IGA, DG formulations can be adopted to handle geometric non-conformities between NURBS patches [START_REF] Zhang | Discontinuous Galerkin methods for Isogeometric Analysis for elliptic equations on surfaces[END_REF]. In particular, Bazilevs et al. [START_REF] Bazilevs | Nurbs-based isogeometric analysis for the computation of flows about rotating components[END_REF] combined a Continuous Galerkin (CG) scheme based on quadratic NURBS with a DG discretization of the sliding interface to solve the incompressible Navier-Stokes equations. An alternative strategy is to employ mortars to deal with the hanging nodes. The mortar approach is based on projections and it was first introduced in the context of spectral element methods to treat non-conforming meshes (13; 14). More recently, the mortar method has been adopted to simulate compressible flows with sliding meshes, combined with compact high-order discretizations (15; 16). Due to the nature of the projections, mortar based methods can suffer from loss of conservativity when dealing with curvilinear sliding interfaces [START_REF] Dürrwächter | An efficient sliding mesh interface method for high-order discontinuous Galerkin schemes[END_REF]. However, Zhang et al. [START_REF] Zhang | A conservative high-order method utilizing dynamic transfinite mortar elements for flow simulations on curved nonconforming sliding meshes[END_REF] demonstrated that a conservative scheme can be obtained by using transfinite mapping to define the mortars. On the other hand, interpolation based DG methods are inherently conservative and significantly less expensive than mortar approaches, as evidenced in [START_REF] Laughton | A comparison of interpolation techniques for nonconformal high-order discontinuous galerkin methods[END_REF].

In this work, we propose an Isogeometric DG method with sliding meshes for compressible flow problems. The NURBS-based representation is exploited to define a watertight interface between the rotating and the fixed subdomain, whereas the DG formulation allows to obtain a fully conservative numerical scheme, suitable for compressible flows. The developed algorithm is explained in details in section 3. The methodology is then verified using a classic benchmark for compressible flows and a thorough error analysis is conducted. Section 5 is dedicated to the investigation of the viscous flow around a pitching ellipse, with the aim of testing the sliding mesh algorithm on a more demanding test case. In particular, a mesh sensitivity analysis is performed, with respect to the grid refinement level, the basis degree and the position of the sliding interface. Moreover, a comparison with a grid deformation approach is presented. We also simulate the pitching ellipse in a supersonic flow condition to demonstrate the ability of the sliding mesh algorithm to deal with shocks. Lastly, in order to prove the potential of the Isogeometric DG framework for more complex geometries, a 3-bladed vertical axis wind turbine configuration is considered in section 6. In particular, the generation of a suitable computational grid is discussed and a flow simulation is carried out. Finally, the main results are outlined in the conclusion and some research perspectives are discussed.

NURBS-based DG with moving meshes

In order to simulate fluid flow problems with moving geometries, we consider the Navier-Stokes equations in ALE form, with a computational domain which moves with a generic velocity V g . The two-dimensional case is presented here, but the proposed approach can be generalized to the three-dimensional case. The divergence form of the equations is the following:

∂W ∂t + ∇ • F c -F v -V g • ∇W = 0, (1)
with W being the vector of conservative variables, F c the convective flux, and F v the viscous flux:

W =      ρ ρu 1 ρu 2 ρe      , F c,i =      ρu i ρu 1 u i + pδ 1i ρu 2 u i + pδ 2i ρu i e + p ρ      , F v,i =      0 τ 1i τ 2i u k τ ki -q i      , ( 2
)
where τ ij is the viscous stress tensor and q i is the thermal conduction flux, defined as:

τ ij = µ ∂u i ∂x j + ∂u j ∂x i - 2 3 µ ∂u k ∂x k δ ij , ( 3
)
q i = -γ µ P r ∂e ∂x i , ( 4
)
where γ = 1.4, P r = 0.72 and µ is determined by the Reynolds number. For inviscid flows, the Euler equations are considered and the viscous flux F v term is ignored.

The second order derivatives introduced by the viscous part of the equations are treated using the Local Discontinuous Galerkin (LDG) approach [START_REF] Cockburn | The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems[END_REF]. To this end, we reduce eq. ( 1) to a system of first order equations by introducing the auxiliary variable G:

     ∂W ∂t + ∇ • F c (W) -∇ • F v (W, G) -V g • ∇W = 0, G -∇W = 0. (5)
In order to obtain a discrete solution, the NURBS-based DG method relies on Rational Bernstein functions(20):

R p i 1 i 2 (ξ, η) = B p i 1 (ξ) B p i 2 (η) ω i 1 i 2 p+1 j 1 =1 p+1 j 2 =1 B p j 1 (ξ) B p j 2 (η) ω j 1 j 2 , ( 6
)
where ξ and η are the parametric coordinates, defined in the parametric domain Ω = [0, 1] 2 . The coefficients ω j 1 j 2 are the weights and B p j are the Bernstein polynomials of degree p. According to the isogeometric paradigm, a unified mathematical representation is adopted for the geometry x and the discrete solution fields w h and g h . Thus, in each element, we can write:

   x w h g h    = (p+1) 2 i=1 R i (ξ, η)    x i w i g i    . ( 7
)
The computational mesh is thus a tessellation of rational Bézier patches, whose geometry is described by the control points x i , whereas w i and g i are respectively the degrees of freedom (DOF) of the conservative variables and their gradients. The eqs. (5) are discretized using the direct ALE method developed in [START_REF] Pezzano | A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes[END_REF], leading to the following weak formulation:

                   d dt w i ˆ Ω R k R i |J Ω j | d Ω = ˆ Ω ∇R k • F c -F v -V g w h |J Ω j | d Ω - ˛∂ Ω R k F * ale -F * v |J Γ j | d Γ, g i ˆ Ω R k R i |J Ω j | d Ω = ˆ Ω ∇R k w h |J Ω j | d Ω - ˛∂ Ω R k W * |J Γ j | d Γ. (8a) (8b)
The convective numerical flux F * ale is computed through a modified HLL Riemann solver [START_REF] Pezzano | A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes[END_REF], whereas the LDG flux [START_REF] Cockburn | The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems[END_REF] is used for F * v and W * . Integration is carried out in the parametric domain Ω using Gauss-Legendre formulas. Due to the mapping between the physical and the parametric space, the Jacobians |J Ω j | and |J Γ j | appear in the integrals. Equation (8b) is solved within each time iteration in a decoupled manner and the system of equations ( 8) can be rewritten as:

d dt Mw = R(w h , V g ), ( 9
)
where M is the mass matrix and the residual R is the right-hand side of eq. (8a). The resulting set of ordinary differential equations (ODE) ( 9) is integrated in time using a 4th-order fourstage explicit Runge-Kutta (RK) scheme. Moreover, it is possible to further simplify the time integration algorithm when the mesh movement is rigid. Indeed, it is trivial to show that the mass matrix is constant in time when the elements are rotating and translating. We thus obtain:

M dw dt = R(w h , V g ). ( 10
)
In this case, the mesh movement contributions are limited to the flux terms due to the grid velocity V g . Integrating equation ( 10) is significantly less expensive from the computational standpoint, as the inverse of the mass matrix is computed in the pre-processing phase. Finally, artificial viscosity terms are added in the case of discontinuous solutions, based on the subcell shock capturing procedure presented in [START_REF] Persson | Sub-Cell Shock Capturing for Discontinuous Galerkin Methods[END_REF] extended to rational representations (23; 24).

Sliding mesh algorithm

In this section we explain in details the treatment of sliding meshes. The proposed approach is presented for 2D problems but can be extended for 3D cases without any modification. The algorithm is characterized by three phases. First, the geometry is updated and the control points of the mesh are moved. As the movement is performed, the connectivity between the interface elements is checked and updated. Once the grid connectivity is established, the interface is subdivided into smaller arcs shared by only two elements. Lastly, a special treatment of the sliding faces is required in order to accurately compute the flux integrals. The mesh is updated before each internal stage of the RK algorithm. Thus, time-consistent residuals are used to solve eq. [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[END_REF]. The three phases are detailed in the following subsections.

Mesh movement and connectivity update

For the sake of simplicity, we analyse the case of a single rotating subdomain Ω r surrounded by a fixed region Ω f . The sliding interface is thus defined as I = ∂Ω r ∩ ∂Ω f . Following the isogeometric approach, the grid velocity field is represented using the same basis adopted for the DG formulation:

V g = (p+1) 2 i=1 R i (ξ, η)v g,i , (11)
where the v g,i is the velocity of the i-th control point x i . Since the mesh velocity is described as a DG field, it is naturally possible to represent discontinuous displacements, which is particularly useful in the context of sliding meshes. Indeed, the sliding interface is characterized by a discontinuity of the tangential velocity V g • t. On the other hand, the normal component V g • n must be continuous, in order to keep the mesh watertight. For perfectly circular interfaces, the control point velocities u g,i , v g,i can be easily computed thanks to the affine invariance property of Bézier patches [START_REF] Piegl | The NURBS Book[END_REF]. Defining the angular velocity ω and the axis of rotation O = (x 0 , y 0 ), we have, for each control point x i , y i in the rotating subdomain Ω r : Assuming that the elements are initially aligned, the discontinuity in the mesh movement generates hanging nodes along the sliding interface, as it can be observed in Fig. 1. Consequently, the degrees of freedom of the elements adjacent to the interface are not aligned. As the inner domain rotates, the position of the hanging nodes changes as well. In order to update the mesh connectivity, we rely on the computation of two angles, as represented in Fig. 2. ∆θ 0 is the angular amplitude of the element edge between x 1 and x 2 , whereas ∆θ s is the angular amplitude of the circular arc between x 1 and the hanging node x s . We then define the ratio s as:

u g,i = -ω(y i -y 0 ), v g,i = ω(x i -x 0 ). ( 12
s = ∆θ s ∆θ 0 . ( 13
)
The current connectivity is valid when 0 < s < 1, whereas it must be updated when the angular ratio s becomes negative or greater than the unity.

Interface splitting

The second step of the sliding algorithm is the subdivision of the interface into a set of elemental faces, which are shared uniquely by two elements. Let us name Ω 0 the rotating element, Ω 1 and Ω 2 the corresponding fixed elements, as illustrated in Fig. 3. The elemental sliding faces are therefore Γ 1 = ∂Ω 0 ∩∂Ω 1 and Γ 2 = ∂Ω 0 ∩∂Ω 2 . We also define Γ

0 = ∂Ω 0 ∩I = Γ 1 ∪ Γ 2 .
Figure 3: Splitting procedure in the parametric and physical spaces

The splitting procedure relies on the properties of the NURBS representation. Indeed, a rational Bézier curve of degree p defined on the parametric interval (0, 1) can be interpreted as a NURBS curve with a the following knot vector:

Ξ = {0, . . . , 0 p+1 , 1, . . . , 1 p+1 }. ( 14
)
Let us define the splitting parameter σ as the value of ξ 0 that corresponds to the physical position of the hanging node x s on the face Γ 0 . It is then possible to divide Γ 0 into the two elemental faces, Γ 1 and Γ 2 , by applying (p + 1)-times the knot insertion algorithm (20) at the parametric coordinate ξ 0 = σ. We thus obtain two Bézier curves with the following knot vectors:

Ξ 1 = {0, . . . , 0 p+1 , σ, . . . , σ p+1 }, Ξ 2 = {σ, . . . , σ p+1 , 1, . . . , 1 p+1 }. (15)
The control points of the new curves are obtained by recursively using p times the knot insertion formula, specialized here for Bézier representations:

x r+1 i = (1 -α r i )x r i-1 + α r i x r i , α r i =        1 if i ≤ r + 1 σ if r + 2 ≤ i ≤ p + 1 0 if i ≥ p + 2 (16)
with r = 0, . . . , (p -1). The result of the splitting procedure is illustrated in Fig. 3 for a quadratic representation.

In order to adopt the described splitting algorithm, the value of σ must be known. Due to the rational nature of the representation, computing σ is not trivial. The procedure that allows to determine the parameter corresponding to a given point of a rational Bézier curve is the point inversion algorithm [START_REF] Piegl | The NURBS Book[END_REF]. If we denote C as the Bézier representation of the Γ 0 curve, we can write:

C(σ) = p+1 i=1 B p i (σ) ω i x i p+1 j=1 B p j (σ) ω j = x s , ( 17
)
where ω i and x i are respectively the weights and the control points of the considered edge. The problem of determining σ is solved by minimizing the square of the distance between the curve C and the hanging node:

g(ξ 0 ) = ||C(ξ 0 ) -x s || 2 . ( 18
)
Imposing g (σ) = 0, one obtains:

f (σ) = C (σ) • C(σ) -x s = 0. ( 19
)
The solution of eq. ( 19) is achieved by means of an iterative algorithm. Piegl et al. [START_REF] Piegl | The NURBS Book[END_REF] adopted the Newton algorithm to solve the point inversion problem. Such an approach requires to analytically compute the first derivative of f , which contains the second derivative of the Bézier curve C. In order to avoid the costly computation of C , we employ the secant algorithm, which results in the following recursive formula:

σ i = σ i-2 f (σ i-1 ) -σ i-1 f (σ i-2 ) f (σ i-1 ) -f (σ i-2 ) , ( 20
)
and we initialise the algorithm taking σ 1 equal to the angular ratio s used to update the connectivity, and σ 0 equal to the splitting parameter computed for the previous time step.

The convergence conditions to stop the secant iterations are:

f (σ i ) ≤ 10 -15 , |σ i -σ i-1 | ≤ 10 -14 . ( 21
)
Since the values used to initialise the algorithm already provide a close estimate of the actual splitting parameter, few iterations are required. In practice, we observed that the convergence conditions are always satisfied in less than 5 iterations.

Flux computation

Once the interface is split into the elemental faces, it is possible to compute the flux integrals of the DG scheme. For each elemental segment of the interface, two integrals have to be computed, one for the rotating element and one for the fixed element. The difficulty of computing the flux integrals is due to the non-matching parameterisation on the two sides of the interface. Indeed, due to the lack of alignment of the elements between Ω r and Ω f , the face Γ 1 is described using two different parameterisations and the same is true for Γ 2 , as presented in Fig. 3. If we focus on Γ 1 , the flux integrals for a numerical flux function F * are:

f - 1 = ˆΓ1 R k ξ 1 F * w - h (ξ 1 ), w + h (η 1 ), n dΓ 1 , ( 22
)
f + 1 = ˆΓ1 R k η 1 F * w - h (ξ 1 ), w + h (η 1 ), n dΓ 1 . ( 23
)
One could naively try to use as quadrature points the Gauss-Legendre nodes on the interval ξ 1 ∈ [0, σ] for f - 1 and on the interval

η 1 ∈ [1 -σ, 1] for f + 1 .
However, the positions of the integration points in the physical space do not match, due to the non-linearity of the coordinate transformation, resulting in a locally non-conservative scheme. The effect of the loss of conservativity is presented in Fig. 4a for a freestream preservation problem. We can clearly observe the spurious oscillations around the sliding interface. We can instead obtain a fully conservative scheme by computing both flux integrals on the parametric interval ξ 1 ∈ [0, σ]. We therefore rewrite f - 1 and f + 1 as:

f - 1 = ˆσ 0 R k ξ 1 F * w - h (ξ 1 ), w + h η 1 (ξ 1 ) , n |J Γ 1 | dξ 1 , ( 24
)
f + 1 = ˆσ 0 R k η 1 (ξ 1 ) F * w - h (ξ 1 ), w + h η 1 (ξ 1 ) , n |J Γ 1 | dξ 1 , ( 25
)
where |J Γ 1 | is the Jacobian of the map between ξ 1 and Γ 1 . In order to approximate expressions [START_REF] Hesthaven | Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications[END_REF] and [START_REF] Duvigneau | CAD-consistent adaptive refinement using a NURBS-based discontinuous Galerkin method[END_REF] with numerical quadrature, the reparameterisation η 1 (ξ 1 ) has to be computed for the quadrature points. If we denote ξ 1 gp as the parametric coordinate of a generic Gauss-Legendre point, then its physical coordinate is equal to:

x gp = p+1 i=1 B p i (ξ 1 gp ) ω i x i p+1 j=1 B p j (ξ 1 gp ) ω j = p+1 i=1 B p i (η 1 gp ) υ i y i p+1 j=1 B p j (η 1 gp ) υ j , ( 26
)
where x i and ω i are the control points and weights of the elemental face, whereas y i and υ i are the control points and weights of the edge of Ω 1 . The reparameterisation η 1 gp is found by solving eq. ( 26) with the point inversion algorithm described in the previous section. The same treatment is then repeated for the integrals over the face Γ 2 :

f - 2 = ˆ1 σ R k ξ 2 F * w - h (ξ 2 ), w + h η 2 (ξ 2 ) , n |J Γ 2 | dξ 2 , ( 27
)
f + 2 = ˆ1 σ R k η 2 (ξ 2 ) F * w - h (ξ 2 ), w + h η 2 (ξ 2 ) , n |J Γ 2 | dξ 2 . ( 28
)
In order to show the effectiveness of the proposed strategy, we report in Fig. 4b the result of the freestream preservation test and compare it with non-conservative integral computation. We can observe that the spurious oscillation around the interface are absent and that the constant solution is well preserved.

Verification: isentropic vortex

The developed sliding mesh technique is firstly tested on an analytic problem governed by the compressible Euler equations. The considered test case consists in the advection of an isentropic vortex, which is a common benchmark for assessing the accuracy of compressible flow solvers [START_REF] Hesthaven | Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications[END_REF]. The time evolution of the flow is described by the following function:

                       ρ = 1 - γ -1 16γπ 2 β 2 e 2(1-r 2 ) 1 γ-1 u = 1 -β y -y 0 2π e 1-r 2 v = β x -t -x 0 2π e 1-r 2 p = ρ γ ( 29
)
where r = (x -t -x 0 ) 2 + (y -y 0 ) 2 , with x 0 = 5, y 0 = 0 and β = 5. The computational domain is [2.5, 10] × [-2.5, 2.5] and the boundary fluxes are computed using the analytic solution ( 29). The computational domain is divided into two regions, a rotating one and a fixed one, as illustrated in Fig. 5. The radius of the sliding interface is equal to 1.5, the centre of rotation of the inner region corresponds to the initial axis of the vortex, and a unitary rotational frequency is adopted. The core of the vortex is therefore inside the rotating subdomain in the initial phase of the simulation. As the vortex is advected downstream, it crosses the sliding interface to enter the fixed subdomain, as it can be observed in Fig. 5b. The proposed test case allows to investigate how well the sliding interface is capable of preserving the flow characteristics.

In order to perform a rigorous error analysis, a convergence study is carried out. Starting from the baseline mesh presented in Fig. 5a, isotropic refinement is applied to obtain four different levels of mesh resolution. Quadratic, cubic and quartic basis functions are tested. For each combination of degree and mesh level, a simulation is performed considering a fixed inner subdomain as well. By comparing the results obtained with the two approaches, it is possible to quantify the error introduced by the computation of the flux integrals on the sliding interface. The L 2 -norm of the error for the total energy field is evaluated at t = 2 using numerical quadrature. The results of the convergence analysis are presented in Fig. 6. It can be observed that optimal convergence rates are obtained for all basis degrees for both fixed and sliding meshes. Furthermore, roughly the same errors are obtained with the two techniques, meaning that the error generated by the sliding interface is negligible and flow characteristics are perfectly preserved.

= ˆΩ w h (T f in ) dΩ -ˆΩ w h (T ini ) dΩ + ˆTfin T ini ˛∂Ω F(w h ) dΓ dt. ( 30
)
The space integrals are computed with the same quadrature rule employed for the DG formulation while the time integral is evaluated by means of the same RK method used to solve the ODE system [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[END_REF]. The coarse mesh illustrated in figure 5a

Case study: pitching ellipse flow

The second test case concerns the bidimensional flow around a pitching ellipse. Thanks to the use of rational Bézier functions, the ellipse can be exactly represented. Thus, there is no additional error introduced by the discretization of the computational domain. The aspect ratio of the ellipse is equal to 12 and its angle of attack evolves in time with the following law:

α(t) = -A sin(2πf t), ( 31
)
with reduced frequency k = πf c/U ∞ = 0.5π and amplitude A = 30 • . The ellipse is pitching about its center. We consider a laminar subsonic flow condition with freestream Mach number equal to 0.2 and a Reynolds number of 500. The flow configuration is characterized by a periodic vortical wake generated by the dynamic separation of the boundary layer around the ellipse. Therefore, the aim of the proposed test case is to study the behaviour of the developed sliding mesh technique for complex unsteady flows.

Mesh convergence

Firstly, a mesh convergence study is conducted. We consider a large computational domain, delimited by the rectangle [-30c, 60c] × [-30c, 30c], with c being the major axis of the ellipse. The far-field boundary conditions are weakly imposed using Riemann invariants. On the ellipse surface, a weakly prescribed no-slip adiabatic wall boundary condition is adopted. Since the isogeometric approach allows an exact representation of the boundary, an extremely coarse baseline mesh of degree two can be generated. As represented in Fig. 7, the meshes adopted for the actual computations are obtained applying a hierarchical refinement to the baseline configuration. The convergence analysis is carried out using 3 mesh refinement levels:

• level 1: 1464 total elements, 16 of which on the ellipse boundary,

• level 2: 2406 total elements, 32 of which on the ellipse boundary,

• level 3: 5046 total elements, 64 of which on the ellipse boundary.

The study is performed using rational basis defined using quadratic, cubic and quartic polynomials. The radius of the sliding interface is equal to 0.75c, meaning that the minimum distance between the ellipse and the interface is of a quarter of the major axis length c.

The initial condition of the simulations corresponds to the uniform freestream flow state. After a numerical transient, the solution converges to a stable limit cycle in which the flow fluctuations are synchronized with the pitching movement of the ellipse. We present in Fig. 8 the evolution of the aerodynamic coefficients over one pitching cycle. We compute, for each cycle, the average drag coefficient Cd , the peak lift coefficient C l and the energy transfer coefficient E. The latter being defined as the area delimited by the C m -α curve:

E = 1 1 2 ρ ∞ u 2 ∞ c 2 ˛Mz dα = ˛Cm dα, ( 32
)
where M z is the moment of the aerodynamic forces with respect to the pitching axis. The simulations are stopped when the considered coefficients are sufficiently converged in time. The results of the mesh convergence analysis are reported in Fig. 9. All the tested configurations of degrees and refinement levels are able to reproduce the synchronization between the flow and the pitching movement. As expected, a significantly faster convergence is observed when the degree of the basis is increased. At the same time, the coarsest mesh is not capable of predicting with accuracy all the coefficients, even with quartic polynomials. Indeed, the level 2 grid with a cubic representation provides a better estimation of Cd , with a similar number of degrees of freedom. Therefore, we conclude that the cubic level 2 mesh represents a satisfactory compromise between accuracy and computational cost. In general, the best trade-off is found by combining h and p refinements. We can also remark that the value of E is negative, which means that, from a physical point of view, the energy is transferred from the ellipse to the fluid.

Comparison with deforming mesh

In the second part of the ellipse case study, we compare the sliding mesh approach with the deformation technique validated in [START_REF] Pezzano | A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes[END_REF]. Since the pitching amplitude is limited to 30 • , it is still possible to continuously deform the computational domain without incurring into an excessive distortion of the mesh, which could potentially decrease the accuracy of the solution.

In order to compare the two techniques, we repeat the convergence study with the grids of Fig. 7 using the ALE methodology. To this end, a mesh velocity field V g needs to be defined. The computational domain is thus divided in three regions: an inner one that oscillates rigidly with the ellipse, an outer fixed zone and a transition region. The velocity of the control point net is computed with the following law:

u g (x, t) = -ω y v g (x, t) = ω x ( 33
)
with ω = α σ(x, t), where σ is a piecewise-defined blending function:

σ(x, t) =            1, if R ≤ R int 1 2 1 + cos π R -R int R ext -R int , if R int < R < R ext 0, if R ≥ R ext (34) with R = x 2 + y 2 .
The distance R int is equal to the radius of the sliding interface and R ext = 4c. Intuitively, the sliding mesh movement can be interpreted as the limit case where R ext = R int . We illustrate in Fig. 10 the differences between the deforming and the sliding meshes. Besides, we can observe that, despite the significant differences between the two grids, the density fields obtained with the two approaches are visually identical.

In order to better compare the two approaches, Cd , C l and E are computed for the deforming mesh as well. The numerical values of the three aerodynamic coefficients are reported in Tables 2, 3 and 4 respectively. We can observe that the same converged values are obtained with both the techniques. Therefore, we conclude that the proposed sliding mesh approach is correctly capable of simulating complex flows. Although some small discrepancies are observed between the values obtained using the level 1 meshes, there is no evidence to infer that one methodology is more accurate than the other. However, the sliding approach is less expensive from a computational point of view, because the mass matrix is constant in time, and is less limited in terms of movement amplitude.

Sliding Deformation

Influence of the sliding interface radius

We then employ the ellipse case study to investigate the effect of the placement of the sliding interface on the numerical results. Low-order sliding mesh approaches can introduce numerical artifacts in the approximate solution, due to a loss of conservativity and poor interpolation. Ideally, the sliding interface should not be in close proximity of the boundary in order to limit the effects of spurious phenomena on the aerodynamic coefficients. On the other hand, a smaller rotating region allows to reduce the computational costs, because the smaller tangential velocity of the mesh allows to increase the time step. Therefore, the generation of meshes with sliding interfaces is the result of a trade-off and the process heavily relies on user experience. In this context, the use of a geometrically exact movement coupled with a high-order interpolation can significantly improve the accuracy of sliding grids and simplify the mesh generation task. The previous convergence analysis was carried out considering a sliding interface radius equal to 0.75c. By slightly modifying the baseline grid 7a, we were able to obtain three additional meshes, with different placements of the interface: one with a smaller radius of 0.625c, and two with a larger rotating region, with r equal to c and 1.5c, as illustrated in Fig. 11. In order to provide a fair comparison, the refinement process has been adapted to obtain a similar resolution in the wall area for all the considered grids. A cubic representation is employed and the overall mesh resolutions are comparable to the level 2 refinement adopted in the convergence study.

The values of Cd , C l and E obtained for the different positions of the sliding interface are reported in Table 5. We do not observe a significant variation of the aerodynamic coefficients with respect to the radius of the interface. One can notice that the most sensitive quantity is the average drag coefficient. However, the relative variation between the maximum and the minimum value is less than 0.13%, which is completely negligible in a practical application. We can therefore state that, thanks to the accuracy of the developed sliding mesh technique, the results are nearly independent with respect to the position of the sliding interface. This simplifies the mesh generation problem, as the subdivision of the computational domain will be primarily dictated by the geometrical features of the boundary.

Supersonic flow

We conclude the ellipse case study with a simulation in the supersonic regime. The amplitude of the pitching motion is reduced to A = 15 • and a freestream Mach number equal to 2 is considered. The computations are performed using the meshes used for the subsonic regime, illustrated in Fig. 7. The simulation of supersonic flows requires robust numerical algorithms, because of the presence of strong shocks. Furthermore, non-conservative schemes can lead to incorrect estimations of the shock velocity. The considered flow is characterised by a bow shock in front of the ellipse. As the angle of attack α oscillates, the shape and intensity of the bow shock evolve. The density fields for three different values of α are illustrated in Fig. 12, for the finest grid with quadratic bases. As we can observe, the discontinuity is well resolved and the sliding interface does not interfere with the movement of the shock, since there is no loss of conservativity. As in the subsonic case, the solution converges to a stable limit cycle after the initial transient. In Fig. 13 we report the limit cycle curves for the aerodynamic coefficients C l , C d and C m , for the same configuration. To validate more in depth this case, a mesh convergence study is achieved for quadratic, cubic and quartic bases. The evolution of the average drag coefficient Cd , the peak lift coefficient C l and the energy transfer coefficient E are reported in Fig. 14. As expected, achieving grid convergence is more challenging in the supersonic regime than in subsonic conditions, due to the necessity to use fine grids to correctly capture the shock and its evolution. Nevertheless, very close values are obtained using the finest grid for all bases. Sliding meshes are commonly employed to simulate rotating machinery such as helicopter rotors [START_REF] Steijl | Sliding mesh algorithm for CFD analysis of helicopter rotor-fuselage aerodynamics[END_REF], axial compressors [START_REF] Gourdain | Prediction of the unsteady turbulent flow in an axial compressor stage. Part 1: Comparison of unsteady RANS and LES with experiments[END_REF] and wind turbines [START_REF] Patil | Large eddy simulation of an H-Darrieus rotor[END_REF]. These kind of applications are characterized by complex geometries and topologies. For this reason the present section is dedicated to the investigation of a more challenging test case. In particular, we consider a Vertical Axis Wind Turbine (VAWT).

Construction of the computational domain

The first challenge of the proposed application is the generation of a suitable high-order mesh to perform the flow simulation. We start by defining the geometry of the turbine. Among the several possible VAWT designs, we consider an H-Darrieus rotor type [START_REF] Daróczy | Comparative analysis of turbulence models for the aerodynamic simulation of H-Darrieus rotors[END_REF] with three NACA 0018 blades. Being N the number of blades, the radius of the turbine R and the chord of the blades airfoils c are defined by means of the solidity σ = N c/R, which is equal to 0.5 for our application. A cylindrical mast of radius r = 0.2c completes the geometry of the rotor. The outer boundaries are sufficiently far from the turbine in order to allow the development of the wake and to avoid unwanted acoustic perturbations due to spurious reflections. As for the ellipse case study, the far-field boundary conditions are based on Riemann invariants and the rotor surfaces are considered to be adiabatic no-slip walls. A rational quadratic representation is adopted for the rotor geometry and the sliding interface. The mast is therefore exactly described, whereas the NACA airfoil is approximated via least square fitting. In order to generate the mesh, we exploit the periodic pattern of the geometry. We first create a very coarse block-structured grid, presented in Fig. 15a. The baseline grid must respect the constraint of having a uniform subdivision of the sliding interface. Moreover, due the scale differences between the mast radius, the blade chord and turbine diameter, the initial patches are visibly irregular. We illustrate in Fig. 15b the mesh obtained after one isotropic refinement step and the Jacobian of the isogeometric map. It can be observed that strong discontinuities in the value of the Jacobian are present between the mesh blocks.

In order to improve the mesh regularity, we adopt a simple elliptic relaxation technique. The developed approach is an adaptation to high-order Bézier grids of the barycentric smoothing algorithm described by Falsafioon et al. [START_REF] Falsafioon | Comparison of Two Mesh Smoothing Techniques for Unstructured Grids[END_REF]. In each step of the relaxation procedure, the updated position of the j-th control point is equal to the barycentre of the neighbouring control points:

x k+1 j = 1 n n i=1 x k i , (35)
with n being the number of neighbours of x j . The sequence defined by eq. ( 35) converges towards the solution of the Laplace equation for the coordinate fields. The mesh illustrated in Fig. 15c is obtained applying the barycentric smoothing algorithm for 50 iterations starting from the grid of Fig. 15b. We notice that the regularity is significantly improved and that the gradient of the Jacobian is globally reduced. The mesh used for the actual computations is obtained by locally refining the smoothed grid in the rotor region, in particular around the mast and the blades, as presented in Fig. 16. The resulting grid consists of 18545 elements in total.

Flow simulation

The typical working conditions of VAWTs are characterized by turbulent flows. Indeed, the blades need to have a sufficiently high lift-to-drag ratio to generate power, meaning that the flow regime of the airfoil has to be at least transitional. Simulating such a flow configuration requires turbulence modelling, high mesh resolution and implicit time integration, which is out of the scope of the current work. For these reasons, the present simulation is only carried out in the laminar regime, instead of considering fully realistic flow conditions. The freestream Mach number is equal to 0.1 to avoid unwanted compressibility effects. The rotational speed ω of the turbine is defined using the tip-speed ratio λ = ωR/u ∞ . For the present simulation λ = 2, meaning that the tangential velocity of the blades is twice the freestream speed. The Reynolds number Re rotor computed using u ∞ and the turbine radius is equal to 3000. In order to estimate the flow regime around the airfoils, we can consider the maximum possible relative velocity of the blades with respect to the freestream condition and compute:

Re blade = c (u ∞ + ωR) ν = σ N (λ + 1) Re rotor = 1500. ( 36
)
The flow around the blades is therefore laminar. In a realistic condition Re blade ≈ 10 5 at least. The result of the laminar regime is the separation of the blades boundary layer very close to the leading edge and the consequent formation of large vortical structures, as it can be observed in Fig. 17. As the large structures are advected, interactions occur between the eddies and both the mast and the blades, resulting in a complex chaotic flow. A total of 10 revolutions are simulated. The evolution of the power coefficient C p and of the aerodynamic force coefficients of the turbine are reported in Fig. 18. Due to the chaotic nature of the solution it is not possible to extract a periodic pattern for the C p curve. Indeed, the interactions between the rotor and the vortical structures generates strong fluctuations in the power coefficient. Also, we can notice that the value of C p is negative, meaning that the movement of the turbine is absorbing energy. This is a direct consequence of the poor aerodynamic performance of the blades in the laminar regime. Nevertheless, this study illustrates the capability of the proposed sliding mesh algorithm to deal with complex configurations.

Conclusion

In this paper we developed a high-order accurate sliding mesh technique for compressible flows. The proposed methodology is based on the isogeometric paradigm: a CAD-consistent DG formulation has been employed to discretize the Navier-Stokes equations. Then, starting from a general ALE framework, we detailed the numerical formulation and the treatment of the sliding interface. In particular, the properties of the NURBS representations were employed to implement an accurate and fully conservative sliding mesh algorithm.

The proposed approach has been firstly verified on a classic benchmark problem for inviscid flows and optimal convergence rates were observed for all the tested basis degrees. Furthermore, the comparison with an equivalent fixed grid showed that the error introduced by the sliding interface is negligible. As a second test case, we studied the viscous flow around a pitching ellipse, and the obtained results confirmed the precision and robustness of the sliding algorithm. We were able to compare the sliding mesh technique with a previously validated ALE formulation based on mesh deformation. We then showed that the outcome of the simulation is independent with respect to the position of the sliding interface. Lastly, we demonstrated the versatility of the developed methodology considering a supersonic flow condition.

As last case study, we simulated the flow around a 3-blade VAWT configuration, allowing us to evaluate the potential of the proposed approach on a more complex geometry. We especially focused on the construction of the computational grid, starting from a set of coarse Bézier patches. In order to obtain a better mesh quality, a smoothing algorithm has been employed, showing that ideas originating from unstructured grid generation can be beneficial in the context of isogeometric analysis as well.

Figure 1 :

 1 Figure 1: Generation of hanging nodes due to the sliding movement

Figure 2 :

 2 Figure 2: Definition of the angles for connectivity update

  (a) Non-conservative interface (b) Fully conservative interface

Figure 4 :

 4 Figure 4: Freestream preservation with non-conservative and conservative sliding interfaces

2 Figure 5 :

 25 Figure 5: Isentropic vortex, setup

Figure 6 :

 6 Figure 6: Isentropic vortex, error analysis

Figure 7 :

 7 Figure 7: Different refinement levels for the ellipse simulation

Figure 8 :

 8 Figure 8: Pitching ellipse, limit cycle solution, cubic level 2 mesh

Figure 9 :

 9 Figure 9: Pitching ellipse, convergence of the aerodynamic coefficients

Figure 10 :

 10 Figure 10: Comparison of sliding and deforming meshes, density field, quartic level 2 mesh

Figure 11 :

 11 Figure 11: Meshes with different sliding interface placements

  (a) α = 0 • , ascending phase (b) α = 15 • (c) α = 7.5 • , descending phase

Figure 12 :

 12 Figure 12: Supersonic pitching ellipse flow, density fields

Figure 13 :

 13 Figure 13: Supersonic pitching ellipse flow, limit cycle solution

Figure 14 :

 14 Figure 14: Pitching ellipse, convergence of the aerodynamic coefficients in supersonic regime

Figure 15 :

 15 Figure 15: Initial VAWT mesh, refinement and smoothing. The colorscale represents the Jacobian of the Isogeometric map.

Figure 16 : 3 Figure 17 :

 16317 Figure 16: Final VAWT mesh

Figure 18 :

 18 Figure 18: VAWT simulation, evolution of the aerodynamic coefficients

Table 1 :

 1 is used for these tests, carried out with both fixed and sliding grids. The resulting conservation errors in mass, momentum and energy are reported in table 1 for different basis degrees. It can be observed that the conservation error is close to the machine precision and does not depend neither on the grid movement nor the basis degree. Conservation error for vortex test-case

			Sliding		Fixed
	Degree	Mass	Mom. x Mom. y Energy	Mass	Mom. x Mom. y Energy
	2	2.7E-13 -6.9E-14 7.5E-16 -1.4E-13 2.8E-13 -3.9E-14 -3.8E-16 6.7E-14
	3	2.2E-13 -1.2Ee-13 -9.1E-15 -2.7E-13 2.2E-13 -1.8E-13 -3.9E-15 -5.0E-14
	4	-1.9E-13 5.9E-14 -1.7E-15 4.5E-14 -3.5E-13 -1.6E-14 -1.2E-15 1.6E-13

Table 2 :

 2 Average drag coefficient for sliding and deforming meshes

	Degree Level 1 Level 2 Level 3 Level 1 Level 2 Level 3
	2	0.4924 0.4846 0.4839 0.4916 0.4833 0.4838
	3	0.4882 0.4837 0.4841 0.4835 0.4836 0.4841
	4	0.4827 0.4840 0.4840 0.4820 0.4840 0.4840
			Sliding			Deformation
	Degree Level 1 Level 2 Level 3 Level 1 Level 2 Level 3
	2	2.694	2.623	2.623	2.691	2.619	2.622
	3	2.645	2.624	2.626	2.630	2.624	2.626
	4	2.627	2.626	2.626	2.624	2.626	2.626

Table 3 :

 3 Peak lift coefficient for sliding and deforming meshes

		Sliding	Deformation
	Degree Level 1 Level 2 Level 3 Level 1 Level 2 Level 3
	2	-0.7207 -0.7070 -0.7055 -0.7204 -0.7059 -0.7054
	3	-0.7114 -0.7053 -0.7055 -0.7065 -0.7052 -0.7055
	4	-0.7054 -0.7054 -0.7054 -0.7047 -0.7054 -0.7054

Table 4 :

 4 Energy transfer coefficient for sliding and deforming meshes

Table 5 :

 5 Variation of the aerodynamic coefficients with respect to r/c

	r/c	0.625	0.75	1.00	1.50
	Cd 0.4834 0.4837 0.4839 0.4840
	C l	2.6246 2.6244 2.6248 2.6247
	E -0.7052 -0.7053 -0.7054 -0.7054

Acknowledgements

The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.

Code Repository

The developed methodology is implemented in the Igloo software suite, which has been employed to perform all the presented computations. The source code and data are available, under the GNU General Public Licence v3, at the following repository: https://gitlab. inria.fr/igloo/igloo/-/wikis/home.