Gabor Lukacs'

Kaitai Shinsho

Geka Sōden

Grégoire Espesset

Traditional Chinese Knowledge before the Japanese Discovery of Western Science in

This review article has quite a long history,

unquestionable erudition on medical and, to some extent, historical matters, but I soon realized, with some disappointment, that Chinese traditional knowledge had not been given a fair treatment by comparison with European science.

Though Lukacs had used-and fully acknowledged as such-my contribution to write most of Part I, Chapter XI of his book, he had never disclosed to me any other chapter at any stage of research or writing, nor did he invite me to participate in the final proofreading. Had he done so, many mistakes could have been detected and corrected before publication. As I read I began to jot down notes and eventually started writing a review. That review-an earlier, much shorter version of the present article-was submitted to Zinbun: Annals of the Institute for Research in Humanities (Kyoto University), and reportedly accepted for publication sometime during 2008, before the editorial committee of that Japanese periodical changed its mind on its next meeting, without offering any explanation. Soon after, I left Europe and lost contact with Lukacs. It was not until very recently that, on the occasion of informal exchanges with Dr. Chang Che-Chia 張哲嘉 (Institute of Modern History, Academia Sinica), my attention was drawn back to Lukacs' book. Chang encouraged me to take up my manuscript, reshape it, and have it published. I hereby gratefully thank him for reading drafts of the present version.

This review article will first describe the contents of the book (section I), then briefly tackle a cluster of form-related issues mainly concerning consistency, typesetting, and copyediting (section II), then move on to problems with transcriptions and translations of Chinese and Sino-Japanese materials (section III), and, last but not least, focus on the treatment of Chinese traditional knowledge betraying ideological biases and outdated value judgments relevant to what may be called the persistence of 'Orientalism' and 'exoticism' in the field of comparative studies (section IV). 1

I

Lukacs' erudite book is the conclusion of nearly a decade of arduous research work. His reader is led through a meticulous bibliographical investigation into one of the earliest phases of Western scientific influence on Eastern Asia, namely the introduction into Japan of European medical knowledge through Dutch works locally adapted into two groundbreaking illustrated manuals of anatomy and surgery. Though the work ultimately bears the name of a single author, the project was made possible by the collaboration of an international network involving at least forty persons, including librarians, scholars and scientists from various fields beyond medicine, booksellers and private collectors, as well as personal friends, as Lukacs gratefully acknowledges in his .

The book is divided into two parts of unequal length, and its contents are structured as units (called "chapters") and sub-units of greatly varying size. The chapter sequence is not always self-evident and abrupt changes of subjects are frequent. Each part is devoted to a manual: the first and longer half (pp. 23-180), to the Kaitai shinsho 解體新書 (A New Book of Anatomy; Edo, 1774), composed in Japanese kanbun 漢文 by Sugita Genpaku 杉田玄白 (1733-1817); the second, shorter half (pp. 181-251), to the Geka sōden 外科宗 傳 (A Complete Manual of Surgery; Nagasaki, 1706), composed in Chinese by Narabayashi Chinzan 楢林鎮山 (1648-1711). 2 The Introduction (pp. 19-22) contains basic elements of historiography and bibliography as well as an overview of the historical backdrop against which the appearance of both manuals into Japan took place. The first part has twelve chapters, divided into about four times as many subchapters. Chapter I (pp. 27-37) introduces the main source of the Kaitai shinsho, namely the Tabulae Anatomicae (Amsterdam, 1731), by Johann Adam Kulmus (1689-1745), an influential textbook on anatomy which was translated into German, Dutch, and French. Leaving Europe behind, Chapter II (pp. 39-48) turns to the Japanese persons involved in the compilation of the Kaitai shinsho, and reviews eleven European medical sources owned and used by them. Chapter III (pp. 49-56) is a detailed description of the cover of the Japanese manual, copied from the frontispiece of Vivae Imagines Partium Corporis Humani (Antwerp, 1566), itself the Dutch edition of the second edition of Juan Valverde's (c. 1520-c. 1588) Historia de la composición del cuerpo humano (Rome, 1559/1560). Chapter IV (pp. 57-65) covers Chinese traditional knowledge and its "taking root" in Japan-this is where Lukacs gets somewhat off-track, as we shall see in my section IV below. In passing are mentioned the writings of two Jesuits-Johann Schreck or Terrentius and Dominique Parrenin (1665-1741)-who tried to "introduce Western anatomy into China" but remained "practically ignored and did not have any influence" (pp. 57-58; both figures return on p. 171). Chapter V (pp. 67-68) introduces very briefly the sources available for the history of the "Rangaku movement" in Japan. 3 Chapter VI (pp. 69-74) focuses on a dissection performed in 1771 by Sugita together with Ogino Gengai 荻野元 凱 (1737-1806), 4 an experiment which lead to Sugita's "unconditional and uncritical adherence to and acceptance of Western anatomy" (p. 74), and discusses the general conditions for performing dissections in Japan from this date up to 1869.

Returning to Europe, Chapter VII (pp. 75-82) tries, but fails, to identify the European person (probably a Dutch surgeon) who brought back from Japan the first copy of the Kaitai shinsho ever to enter the Western world, a copy which was eventually to be purchased by the French Bibliothèque Nationale (Paris) in 1839. Concerned with Japan anew, Chapter VIII (pp. 83-88) presents a tentative printing history of a selection of copies of the Kaitai shinsho, a publication venture which unexpectedly turned out to be a success with 2,000 to 2,500 copies (p. 83; "about seven to ten reprints," we read later on, p. 172) produced in Japan from 1774 onwards-all of which, following the Japanese custom of the time, bear the unaltered date of 1774. Chapter gives the estimated printing date-late 1774 or early 1775-of thirteen copies of the manual containing 4 pages of advertisement inserted by booksellers. Chapter X (pp. 95-106) unfolds an inventory, with descriptive notes, of fifty-five copies from (mostly public) collections in Japan, France, England, Germany, and the United States. The longest and arguably the core-chapter in this first part of the book, Chapter XI (pp. 107-164) is devoted to a detailed study and translation of selected manuscript annotations, authored by past owners, from eight copies of the Kaitai shinsho, a majority of which occur in the book's preface and introduction. Interestingly, only some of these unique annotations directly concern anatomy: others refer to the early development of 'Dutch studies', or allude to, cite, or quote various Chinese classics, or compare chronology and the calendar in East Asia and Europe, and so on. Some long annotations even appear verbatim on different extant copies, testifying that their impact was as strong as that of the text itself-a "faithful word-for-word transfer of notes" also commonly observed in the West (p. 129). This is also where (p. 107) we are belatedly given the original Japanese title and an English translation of Kaitai yakuzu 解體約圖 (Concise Chart of Anatomy), already mentioned on pp. 87 and 90-92, a "short report about the structure of the human body" also by Sugita, which preceded the publication of the Kaitai shinsho proper and convinced Sugita that the "risky enterprise" of having his revolutionary work published could be carried on without fear of censorship and retaliation on the part of the Japanese authorities. Closing the first part of the book, Chapter XII (pp. 165-180) tries to evaluate the necessary; both terms could very well be simply translated as 'Dutch studies' and 'those engaged in Dutch studies' respectively. 4 The Japanese characters for the name Ogino Gengai are not given in Lukacs' book, as well as those for Irako Mitsuaki (see below).

impact of the Kaitai shinsho on Japanese medicine, first in the general context of "disastrous historical and economic events" (p. 166) which immediately followed its publication. In addition to the handwritten materials examined in the preceding chapter, which seem to be unparalleled in pre-modern Japanese medical literature, references to the Kaitai shinsho in other sources-such as minutely detailed illustrated reports of dissection-tend to confirm its immediate influence. Its medium-term impact, however, seems to have been quite limited, considering the fact that a majority of Japanese medical practitioners still openly adhered to Chinese traditional knowledge during the nineteenth century. Only in the waning years of that century did the Japanese government reject Chinese medicine and begin to support Western medical knowledge exclusively.

Devoted to the Geka sōden, a work not found in any public collection in Europe according to Lukcas (p. 219), the second part of the book has thirteen chapters, not subdivided except for chapter VIII (which has thirteen subchapters numbered VIII.-1 to VIII.-4.9). Beginning in Europe, Chapter I (pp. 185-187) introduces the French surgeon Ambroise Paré (1510-1590), who acted as royal surgeon to four consecutive French kings, and his collected works (OEuvres) in 26 volumes, first printed in 1575 and subsequently reprinted several times. Chapter introduces the Dutch edition of Paré's collected works, last printed in 1625, and offers basic information on Dutch surgical knowledge in the seventeenth century.

Moving to Japan, Chapter deals with the introduction of the Dutch edition of Paré's OEuvres into this country, in all likelihood before the end of the seventeenth century, then presents a few Japanese works on surgery "not unlike their Chinese model" (p. 193)-but what this "Chinese model" was remains unspecified. The earliest work listed by Lukacs in this inventory is the Geka sōden, first published in 1706-nearly seven decades before the publication of the Kaitai shinsho. introduce the author of the Geka sōden, Narabayashi Chinzan, an interpreter turned physician, and compare his manual to Paré's collected works. Like Kanbara Horishi before him (1992), Lukacs offers evidence that, for the illustrations of the chapter on the "treatments of wounds," Narabayashi drew inspiration from the Armamentarium Chirurgicum (Ulm, 1655) by Johannes Schultes (1595-1645), alias Scultetus, a German surgeon, rather than from the OEuvres of Paré, whose name does not even appear in the manual. Consisting of the long " , Chapter VI identifies the probable European sources of the illustrations related to the treatment of wounds in Geka sōden, with an appended series of juxtaposed illustrations exemplifying graphic similarities between the former and latter works (pp. 208-213). Similarly, Chapter VII (pp. 215-217) now compares the general structure and selected features from the first part of Geka sōden with its known Chinese source, Chen Shigong's 陳實 功 (1555-1636) Waike zhengzong 外科正宗 (Principles of Surgery)-or, in Japanese reading, Chin Jikko's Geka seisō-dated 1617. 5 The apex of the second part of the book, Chapter VIII (pp. 219-234) describes the structure and contents of a dozen hand-written sources, including the Geka sōden proper and more or less complete manuscripts bearing different titles but believed to be derived from it. Lukacs unfolds long tables proposing reverse identifications of the original Latin names of Narabayashi's Japanese transcription of medicines ("Table II"; see also my section III below) and various oils, ointments and plasters ("Table III").

Chapter IX (pp. 235-239) consists mainly of a translation of a section on "wounds caused by sharp-edged weapons (swords)" (see also my section III) in the Geka kunmō zui 外科訓蒙図彙 (Illustrated Initiation to Surgery;6 Kyoto: 1769; preface dated 1767), by Irako Mitsuaki or Kōken 伊朗子光顕 (1737-1798), a Japanese translation of one of Geka sōden's chapters. Chapter X (pp. 240-244) returns to the Geka sōden and offers some evidence that the textual sources of the chapter on "head wounds" therein must have included Scultetus' Armamentarium Chirurgicum. Chapter XI (pp. 245-248) attempts at socially and historically contextualizing Narabayashi's workof which about fifty manuscript copies may have been produced in Japanand stresses the difficulty to estimate its impact due to the lack of firsthand reports on how contemporaries reacted to its publication. The short Chapters XII (pp. 249-250) and XIII (p. 251) provide further remarks on the influence of Ambroise Paré on Japanese medicine up to the twentieth century. The book has no conclusion.

An appendix on "The Japanese chronological system" (pp. 253-254) gives the correspondence between Japanese reign-periods and the Western calendar, from 1570 to 1868 (see also my section IV below). The "Selected Bibliography" (pp. 255-278) brings together primary and secondary sources in Western and Eastern languages, omitting all Chinese and/or Japanese characters. An Index (pp. 279-286) closes the volume.

II

Though the general appearance of the book suggests at first glance that its layout has been produced with extreme care, minor defects somewhat impede reading pleasure. The general format for transcriptions, English translations, and the author's subsequent personal comments is not fixed, even within a given chapter. Paragraph opening indents seem to have been randomly distributed; many are simply missing while those following empty lines appear to be unnecessary. Numerous typos betray hasty proofreading: "B.C. 202" instead of 206 for the opening year of the Han 漢 dynasty (p. 57); "a unsigned print" (p. 61); "more of less" (p. 167); "haemmorhage" (p. 243); etc. Notes often appear both as marginalia and footnotes, tending to saturate page space, and their numbering is not strictly linear. The reader sometimes must browse through pages to locate a given note, and one note at least appears to be missing (n. 53, p. 233). Another poor authorial or editorial choice was to restrict the use of italics to figure captions and a few subtitles within the structure of the sole main text, which is all the more awkward in a bibliographic work featuring occurrences of book titles on virtually every page, as well as numerous Romanized Sino-Japanese and Chinese phrases; keeping to an example, though the sentence "both Kaitai Hatsumō and Kaizō Zufu copied several illustrations from Kaitai shinsho" (p. 167, original capitalization) seems at first to be about two persons borrowing visual material from a book: it is in fact about the same deed performed by two books.

In addition to forty-six beautiful plates, mostly in color, the book offers a large selection of transcriptions from primary sources. Some of these materials are fully transcribed, then translated, while others are simply translated, without any transcription-which criterion, if any, presided over the insertion or omission of transcriptions is unclear. For example, some readers will find it frustrating not to be allowed to compare the long annotated translation found on pp. 136-137 with its source, a 15-column, 410-word text, according to Lukacs, while a different, much shorter version of the same text is fully transcribed on the ensuing page (p. 138).

Whatever the culture and era under consideration, handwritten Sinitic words often assume graphic variants, which may be difficult to decipher and are occasionally missing from the fonts available on modern computers. For this reason, special care should be taken when typesetting, then proofreading a document containing transcriptions of such words. In Lukacs' book, the word 鲁, which is a common graphic simplification for 魯 (Chin. lu), was wrongly typeset as "曽" twice (p. 125, item 1, b, column 6; item 5, a, column 2); the word "日" (Chin. ri) was misprinted for "曰" (Chin. yue), a very common mistake in ancient as well as modern documents (p. 141, Leaf 1V, 2, a); and the penultimate word in the Japanese title Geka sōden, 傳, was wrongly typed "傅" (p. 192), another timeless error.

As far as Chinese names and book titles are concerned, the official Chinese pinyin transcription system was adopted by the author or the publisher, and yet it has not been applied consistently, as witnessed by a few isolated Wade-Giles transcriptions: "Wu-li hsiao-chih" 物理小識 (p. 134, n. 25), which should read Wuli xiaozhi, and "Chih-fang wai chi" 職方外紀 (p. 149), which should read Zhifang waiji.

III

As the above summary of the book contents has tried to make clear, the main originality of Lukacs' bibliographical investigation is its focus on a corpus of marginalia written by past owners of copies of the Kaitai shinsho. Consequently, the heuristic impact of the work depends extensively on the accuracy of the transcriptions and translations provided for these annotations. In this regard, consistency in translation choices should have been one of the basic rules to follow. A good example of such fluctuating translations is offered by the disyllable 金瘡 (Jap. kinsō). The compound, which I would translate simply as 'puncture wound,' is alternatively rendered as "wounds" (p. 219), arguably too vague; as "war wounds" (p. 232), over-interpretative; and even as the phrase "wounds by sharp-edged weapons" (p. 233), not inadequate perhaps but too long, just like another phrase, "wounds caused by sharp-edged weapons," used to translate the homophonous and synonymous disyllable 金創 from the Geka kinmō zui (p. 235). Another disyllable, 脫肛 (Jap. dakkō), appears twice in the list of the 36 chapter titles from the Geka sōden (items no. 16 and no. 22); first translated as "Gangrene," it becomes "Prolapse of the rectum" (p. 216), suggesting that the transcription of item no. 16 has been corrupted-and perhaps other items in the list.

Several groups of textual occurrences from a beautiful manuscript in Chinese of a Lunyu xu 論語序 (Preface to the Analects of Confucius), reproduced p. 124 (fig. 16), are transcribed and translated on p. 125. Four deciphering problems remain in Lukacs' book, marked by the substitution of a question mark for each unidentified Chinese word. The common (Chinese) source of these occurrences, the Hanshu 漢書 (History of the Han), 7 could have been used to collate the manuscript and easily fill these four gaps:

-3, c, column 4: The missing word in the Chinese official title is 𣴎, a graphic variant for 承, here used for cheng 丞.

-4, e, column 4: The missing word in the Chinese patronymic is 贤, a graphic simplification for xian 賢. 'Counsellor-in-chief Wei Xian' 丞相韋賢 is indeed mentioned in the Hanshu, chap. 30, p. 1717, chap. 80, p. 3311, and chap. 88, p. 3618. 8 -4, c, column 4: The missing word is hou 侯. Xiahou is a well-known ancient Chinese family name, and the earliest mention of a Xiahou Sheng 7 This is the main Chinese historiographical source for the years 206 BC to 24 AD. Its core chapters were completed by Ban Gu 班固 (32-92 AD), after whose death additional materials-in particular some of the monographs or thematic treatises-were completed under the editorship of his sister Ban Zhao 班昭 (49-116 AD).

8 For the meaning of the title chengxiang 丞相, see [START_REF] Hucker | A Dictionary of Official Titles in Imperial China[END_REF], p. 126, no. 483.

following the title 'Great Mentor of the Heir Apparent' 太子太傅夏侯勝 is to be found in the Hanshu, chap. 74, p. 3144.9 -4, g, column 7: The puzzling glyph, combining the radical 歹 with 即, would at first sight appear to be an incorrectly written qing 卿. However, just as our manuscript does, the Hanshu mentions the 'Prince Yang of Langye' 琅邪王陽 together with a 'Yong Sheng, from Jiaodong' 膠東庸生 (Hanshu,chap. 81,p. 3348). Consequently, the mysterious word should be read as yang 陽.

Inaccuracies in the translation of the 118-word Sino-Japanese text given on p. 138 have led the author to a few misunderstandings. To begin with the first line, two words (其後, literally 'afterwards') linking chronologically the first sentence to the next remain untranslated. From this disruption in the narrative sequence of the source proceeds Lukacs' suspicion that the unnamed annotator made a "surprising confusion" (see p. 138, n. a). In fact, rather than confusing persons, the annotator reported two theories concerning the foundation of 'Dutch studies' in Japan: the first theory names as the founder Mito Seizan-kō 水戸西山公, whom Lukacs identifies as Tokugawa Mitsukuni 徳川光圀 (1628-1700), while, according to the annotator, contemporary Japanese scholars in Dutch studies would rather regard Aoki Kon'yō 青木昆陽 (1695-1769) as the founder of their discipline.

The third sentence states that Arai Hakuseki 荒井白石 (1657-1725) 'resumed the study of this knowledge,' as one might translate the phrase 復 修此學, which is inaccurately rendered in the book as "that knowledge was taught to Arai Hakuseki." (In the transcription, 'Hakuseki' appears once as "白石," once as "百石," but it is not stated whether this variation exists in the original handwritten annotation or not.)

Instead of the translation of the fourth sentence, we read, between parentheses: "the meaning of the following twelve characters, underlined in the above text, remained ambiguous and unclear to us." Indeed, thirteen words-not twelve-are underlined in the transcription of the Sino-Japanese text; but the word 島 (Jap. shima) from the place name Tanegashima 種 子 島 should not be included in the so-called unclear passage. The remaining twelve words ( 年十九已 10 通 其國 學 十 八 等 官 , corresponding to the underlined section in the translation below), together with the next sentence, may be translated thus: [The newly disembarked Giovanni Sidotti,] 'aged nineteen, was already well versed in his national studies. 11 An eighteenth-rank official ordered Hakuseki to investigate the truth of the matter'. (Rather than "Hakuseki interrogated (the foreigner).")

In the seventh sentence, the phrase rendered as "very clear" is more evocative in the original text: 'as if he pointed at [the world] in the palm of his hand' 如指諸掌, a formula traditionally regarded as a literary allusion to the Lunyu. 12 The eighth sentence, simply rendered as "Kentoku-in ordered Aoki Kon'yō to translate Rangaku," conveys more details in the original text, which reports clearly that Aoki Kon'yō was ordered 'to become a Translator official and to study Dutch learning' 就譯官學蘭學.

The last chapters of the Blondelet copy of Geka sōden (see p. 223, end of Table II) deal with categories of "Chinese medicines" called "essence" 露水 (Jap. rosui), literally 'dew,' "herbal water" 草水 (Jap. sōsui), and "honey" 蜜 (Jap. mitsu). The first two categories in all likelihood refer to pharmaceutical distillates or macerations of botanical origin, the third one to honeybased syrups. A list of fifteen items in Sino-Japanese-three chapter titles and twelve medicine names-was transcribed by Lukacs, but only the first item in the list has been translated ("method of preparation of rose-water"). The entire series could have been rendered as follows:

Further transcription and/or translation problems include:

-P. 143: Since the names "Ma Xuantai" and "Hua Boren" in the translation do not exist in the original annotation, both should be marked (typically between parentheses or square brackets) as being inserted by the translator.

-P. 154: In the translation of Leaf 8V ("male genital organs"), the word missing from the Japanese phonetic transcription of Steven Blankaart's (1650-1702) name (" 蒲 郎 加 □ 都 ") is 爾 . Here this word is naturally homophonous with 児, a graphic variant for the graph 兒 used in a variant transcription of the same name, 蒲郎加児都 (mentioned p. 155).

-P. 243: The sentence '必可有血走, 不可驚恐哉' has been misinterpreted as a warning about "the danger" of hemorrhage. It should rather be seen as an indication that bleeding is very likely to occur, but that there is no need to be afraid if it does.

IV

The preceding sections already suggest that, when Lukacs did not restrict himself to factual information relevant to modern Western medicine, including anatomy and surgery, his mastery of the contents of the admirable books and manuscripts he has found and, in some cases, purchased in various locations around the world, remained limited, hence his heavy reliance on the transcriptions and translations performed by his international collaborators and Japanese friends. Yet far more prejudicial to the overall quality of the work than these technical shortcomings are Lukacs' obvious clumsiness and passé ideological biases when attempting to position European knowledge in a non-Western setting. The few following examples will illustrate how Lukacs ultimately failed to conceive of a comparative methodology which would have brought into historical perspective the metaphoric encounter of two major paradigms unprepared to confront each other.

The concept of 年號 (Chin. nianhao; Jap. nengō), obscurely rendered as "period of era" (?) on p. 140, remains poorly explained to the lay reader in the main text until a welcome appendix. This appendix, however, could have added that the nianhao system originated in China, centuries before the Japanese nengō system started in 645 AD. Nianhao/nengō is the name of an era sometimes coinciding with a sovereign's whole reign, sometimes with a part of the reign only. Back to the translation on pp. 139-140, not only is the sentence "the second year of Genshi or Kanheitei (in Japanese reading) in China" unclear, it is also misleading. For if 'Genshi' is indeed the Japanese reading of a Chinese reign-period, Yuanshi 元始, which in traditional Chinese chronology roughly corresponds to the first five years AD, 'Kanheitei,' however, introduced in the translation as if it were an alternative name for Yuanshi, is in fact the Japanese reading of the phrase 漢平帝 (Ch. Han Pingdi), literally 'Emperor Ping of the Han [dynasty]'. Indeed the reign of this Chinese emperor did coincide with the Yuanshi (or Genshi) reign-period, but there is otherwise no semantic, lexical, or linguistic identity between both occurrences.

The transcultural approach championed by the author would have required a balanced knowledge of the compared phenomena and, at the very least, an equally fair treatment of them. Regrettably, both occasionally vanish from the present work, especially when the author-a prominent European scientist with an exclusively Western background-comes face to face with some traditional features of Chinese culture. This becomes truly manifest in Chapter IV, "Anatomical knowledge in Japan before the Western influence," where the author attempts to comment on a wellknown Chinese document reproduced on p. 59 (fig. 9), the Neijing tu 內景圖 (Illustration of the Inner Landscape)-a title not translated by Lukacs-and on a Japanese document more or less derived from it, the Inshoku yōjō kagami 飲食養生鑒 (Mirror of Drinking and Eating to Keep Maintaining Life), reproduced on p. 62 (fig. 10). 16 By singling out every Western element lacking on both pictures (such as nerves, lymphatic vessels, and so forth) and pointing out all the "fanciful"-as the author writes twice, about the shape of the bones (p. 58) and the depiction of the pericardium (p. 60)graphic renditions therein from his anachronistic and retrospective modern vantage point, Lukacs obliterates the symbolical and philosophical dimension of a unique, centuries-old knowledge. Beside conventional references to a few secondary sources, which one wonders if the author ever bothered to read, a better understanding of this knowledge might have enabled him to add substance to those critical pages, rather than merely unfolding a descriptive catalogue of what he cursorily calls elsewhere "mistaken ideas about the constitution of the human body" (Introduction,p. 20).

Elsewhere, rather than an English translation, Lukacs chose to render the Chinese phrase 內對丹田 from the Neijing tu (fig. 9, item 26) by a question mark (p. 60). This again may strike the reader as betraying a lack of interest in non-Western knowledge on the part of the author, or perhaps his unwillingness, if not inability, to deal with notions not strictly materialistic and mechanistic. The neglected phrase simply means: 'Inside [the body], facing [this point, is located] the Cinnabar Field.' A fundamental concept in traditional Chinese physiology, abundantly documented in primary sources and studied by modern scholars, Cinnabar Field 丹田 (Chin. dantian; Jap. tanden) also played an important role in various psychophysiological practices such as meditation, visualization, and breathing techniques. Three such Cinnabar Fields came to be located within the human body-respectively in the abdomen, the heart, and the brain, with possible variants-but the fact that they were "devoid of material counterparts" has long been acknowledged by contemporary scholars in the field of Chinese studies. 17 In the same passage, we are told that "nerves were totally unknown" (p. 58) or again that "the existence of nerves was not even suspected in Chinese medicine" (p. 58, n. 21). Then, further on, we meet the disyllable 經 絡 (Chin. jingluo; Jap. keiraku), used by the Japanese to translate into their own language the newly imported European concept (p. 224). Firstly, 'nervous system' would be more accurate than "nerves," given that 'nerve' was translated by yet another disyllable, 神經 (Chin. shenjing; Jap. shinkei), as Dr. Alain Briot informs the reader in his Preface (p. 13). Secondly, the fact the disyllable jingluo/keiraku has had a long history before being used to translate a European word into Japanese is totally ignored by Lukacs. In traditional Chinese physiology and medicine, it refers to an energetic system of 'main conduits' or 'cardinal tracts' (經 jing/kei, often called, albeit improperly, "meridians") and 'network vessels' or 'reticular tracts' ( 絡 luo/raku) linking the external (or superficial) regions of the body to its internal parts. 18 The system is attested to in ancient Chinese medical sources dating back to nearly two thousand years before Japanese physicians ever heard of European physiology. It may be added that this energetic network is also an integral part of the theoretical background of acupuncture, a probably very ancient medical art still practiced inside and outside Asian communities worldwide. 19 Had this pre-modern East-Asian knowledge not been disregarded, the author would easily have completed translation of sentences like "various types of pains resulting from wounds affecting the muscles, the bones, causing paralysis and involving 寒溫 (?)" (p. 224). Here, the problematic compound (Chin. hanwen; Jap. kan'on) may be literally translated as 'cold and warm'. In the Chinese medical tradition, these words designate pathogenic factors, respectively 'cold pathogens'-provoking disorder characterized by contractions, stagnations, chills, and so on-and 'warm pathogens'-responsible for diseases characterized by inflammation, fever, thirst, and the like. Naturally, this binary theory is fully compatible with the 陰陽 (Chin. yinyang; Jap. in'yō) theory, pathogenic cold being viewed as a product of excessive yin activity and pathogenic warmth as an effect of 17 Quoting p. 302 of [START_REF] Pregadio | Dantian 丹田: Cinnabar Field(s); Field(s) of the Elixir[END_REF]. See also the classic essay by Henri Maspero, "Les procédés de 'nourrir le principe vital' dans la religion taoïste ancienne," reprinted in Maspero (1971), pp. 479-589. 18 For discussions of the corresponding Western terminology, see Sivin (1987), pp. 133-147; Hsu (1999), p. 172, n. 22, p. 252. 19 See [START_REF] Unschuld | Medicine in China: A History of Ideas[END_REF], pp. 73-83 ("Structure and Function of the Organism") and 92-99 ("Classic Acupuncture: Origins and Therapeutic Principles").

excessive yang activity. (To be fair, yin and yang are briefly dealt with in notes 43-45, p. 72.) Additionally, the same disyllable 寒溫 may also refer to the 'cold-warm syndrome,' a complex morbid condition combining pathologic coldness and warmth occurring simultaneously in distinct parts of the body. Already during the first century AD, Wang Chong 王充 (27-c. 100) in his Lunheng 論衡 (Doctrines Evaluated) critically discussed the validity of the cold/warm dialectics in a broader correlative context, including physiology and medicine, but also cosmology, politics, and metaphysics. 20 Finally, still in the Chinese medical tradition, the trisyllable 保生湯 (Chin. baosheng tang; Jap. hoseitō), tagged "not clear" by the author and not translated (p. 238), designates a medicinal preparation for parturient women. Its name means, literally, 'decoction to preserve life.' A dozen different recipes for this preparation are attested to in various sources, among which Chen Ziming's 陳自明 (fl. 1237-1271) Furen daquan liangfang 婦 人 大 全 良 方 (Compendium of Effective Prescriptions for Women), sometimes dubbed the most famous medical work of its time, dated 1237centuries before the era covered by Lukacs' book. 21 * Due less to its minor flaws than to the disciplinary limitations of its author and his regrettable Europe-centered viewpoint, Gabor Lukacs' Kaitai Shinsho & Geka Sōden will perhaps not be remembered as a milestone in bibliographic approaches to the comparative history of scientific traditions East and West. The book suffers from two types of imbalance: First, from an imbalanced assessment of two cultures. Nowhere does the author point in detail to some of the numberless "errors" and "fanciful" ideas of Western medicine as he does in the case of the Chinese medical tradition. In all, Lukacs alludes-once and without elaborating-to "all those experienced, well-prepared Western anatomists who kept recording errors for centuries" (p. 72) and concedes that "pre-Vesalian Western anatomy was not distinctively superior to its Eastern counterparts" (p. 191). 22 In two notes, he remarks that, "before 1510, very little reflected scientific truth in Western anatomical treatises" (p. 73, n. 57) and that, "except for surgery and the fight against the mortal effects of smallpox, Western medicine was 20 English translation by [START_REF] Forke | Lun-Hêng, Part I: Philosophical Essays of Wang Ch'ung[END_REF], Chapter XXI, "On Heat and Cold (Hanwên)," pp. 278-284. 21 Various editions of this important source are mentioned in [START_REF] Ng | Male Brushstrokes and Female Touch: Medical Writings on Childbirth in Imperial China[END_REF]; the work itself is analyzed in Chapter Three, "Envisioning Women's Medicine: Chen Ziming," pp. 164-235. 22 The adjective 'Vesalian' refers to Andreas Vesalius (1514-1564), author of the influential Suorum de Humani Corporis Fabrica Librorum Epitome (Basel, 1543).

as helpless in the early nineteenth century against most diseases as its Sino-Japanese counterpart" (p. 74, n. 69). By and large, compared with the Chinese medical tradition and Asian medical figures, Western medical knowledge and Western scientists are given a positive image throughout the book. Second, the book suffers from an imbalanced structure between its two parts and their constitutive chapters-the shortest, in particular single-and two-page ones, should have been combined into larger units. Merging both parts into a single one and following chronology consistently would have benefited the overall clarity and strength of the argument. 23 But did Lukacs try to show anything at all in this book, apart from beautiful pictures of rare manuscripts interspersed with transcriptions and translations produced for him by his collaborators and friends, and cliché developments on the supposed superiority of early modern European minds over their retarded Asian counterpart? There is hardly any substantial discovery in the entire book, apart from limited elaborations on results reached by earlier scholars (acknowledged) and a few hypotheses, and the author must often confess that "further research" would be needed (pp. 78, 82, 105) or that "data" are "lacking" (pp. 129, 191) to reach any conclusion. What remains is a mainly descriptive, richly illustrated album, pervaded with amateurish enthusiasm and the patronizing naïveté of our grandfathers marveling at exotic otherness ("the legendary politeness of the Japanese people," p. 121). Outside of academic milieus, the book may provide the educated public with inspirational reading-some first-person narrative chapters unfold like a bibliophilistic detective novel. In view of its visual quality, it is likely to become a collector's piece. But it is to be feared that it will fuel the essentialist views of less educated readers.

I use the former term as defined by Edward W.Said (1935Said ( -2003) ) with regard to nineteenth-century English literature in his controversial Orientalism(1978), and the latter as studied by[START_REF] White | Nineteenth and Twentieth Century French Exoticism: Pierre Loti, Louis-Ferdinand Céline[END_REF], especially in Chapter I, "Nineteenth and Twentieth Century Traditions and Inventions-A Knowledge Becoming," pp.1-58.

Unless otherwise stated, translated titles of Japanese medical sources are those given by Lukacs in his book while those of Chinese sources are mine.

Incidentally, using the (well established apparently) Western neologisms "Rangaku" (which is a Romanization of the Japanese reading of the words 蘭學) and "Rangakusha" (likewise a Romanization of 蘭學者) does not seem absolutely

The title of this work is not translated in Lukacs' book.

This translation of the title (p. 235) is of course more accurate than that quoted from Janette Doe on the same page ("Secret instruction for surgery"), with which Lukacs rightly disagrees, and than "Illustrations for the Practice of Surgery" given earlier by Lukacs himself (p. 171).

For the translation of the title taizi taifu 太子太傅, seeHucker (1985), p. 485, no. 6256.

Lukacs' transcription, "巳", is here certainly erroneous, even if a common variant.

On the Neijing tu, see[START_REF] Despeux | ). A concluding part, preferably more thoroughly researched than the corresponding current units, would have discussed how both works impacted pre-modern Japan. As for the[END_REF].


Ignacio Durán
email: duran@cerfacs.fr

Stéphane Moreau

Thierry Poinsot

Analytical and numerical study of direct and indirect combustion noise through a subsonic nozzle

. Previous studies have addressed the case where the EWG nozzle is choked and have demonstrated that indirect noise was large compared to direct noise, suggesting that indirect noise only could be retained for gas turbine studies. In the present study, subsonic cases (where the nozzle is unchoked) are analysed using first a full numerical resolution of the unsteady Euler equations, second an analytical method based on the work of Marble and Candel [2] in the low frequency limit and finally the one-dimensional linearized Euler equations in the frequency domain. Results show that direct noise cannot be neglected in these situations and will have to be included for real gas turbines where the flow remains

Noise emissions are a major issue for aircraft and engines manufacturers due to the proximity of airports to residential zones and to the increasing international noise regulations defined by the ICAO. During the last decades, research efforts allowed a significant reduction of jet, fan and external aerodynamic noise which must be continued in order to meet future regulations. The reduction of these sources has increased the relative influence of other noise sources in the aircraft.

For these sources the possible reduction is relatively high, as physical mechanisms governing them

are not yet fully understood. One of these sources is combustion noise, which is generated by heat fluctuations induced by the turbulent flame. These fluctuations generate acoustic but also entropy waves that propagate through the turbine stages to reach the outlet of the engine, contributing to the global engine noise.

Two mechanisms of combustion noise generation can be identified [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF]: direct combustion noise, which is due to acoustic waves created in the combustion chamber that propagate through the turbine stages, and indirect noise, which appears when entropy waves (hot spots) are accelerated through the turbine. The acceleration of these entropy waves generates pressure fluctuations that propagate upstream (contributing to combustion instabilities when a positive feedback occurs), and downstream from the turbine stage (generating indirect combustion noise, as shown in [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF][START_REF] Howe | Indirect combustion noise[END_REF]). Both mechanisms of combustion noise generation were identified by Marble and Candel [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] using the linearized Euler Equations (LEE) and an analytical method to solve the propagation 4 equations considering a quasi-1D configuration, based on the compact nozzle hypothesis. This hypothesis was used to extend the analytical method to a 2D configuration of a turbine stage (Cumpsty and Marble [START_REF] Cumpsty | The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise[END_REF]). The study of combustion noise is directly linked to the propagation of acoustic and entropy waves through the turbine stages, where the mean flow is accelerated and decelerated repeatedly. While the propagation of noise in unconfined domains has been studied

in depth [START_REF] Lighthill | On sound generated aerodynamically. i. general theory[END_REF][START_REF] Colonius | The sound generated by a two-dimensional shear layer: The far field directivity from computations and acoustic analogies[END_REF][START_REF] Goldstein | A generalized acoustic analogy[END_REF][START_REF] Bailly | Modelling of sound generation by turbulent reacting flows[END_REF], no acoustic analogy exists for the propagation of acoustic and entropy waves through turbo-machinery. One and two-dimensional models of wave propagation must be used to propagate the noise generated in the combustor to the outlet of the engine. Waves generated in the combustion chamber can be simulated using advanced Large Eddy Simulations (LES) and the noise at the outlet can be predicted by combining them with these analytical models of waves propagation [START_REF] Cumpsty | The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise[END_REF]. This can be done without having to combine LES results and a turbine stage to simulate the combustion chamber, which would be extremely CPU demanding. Leyko et al. [START_REF] Leyko | Comparison of direct and indirect combustion noise mechanisms in a model combustor[END_REF] used one-dimensional analytical methods combined with numerical tools to show that the ratio of indirect to direct combustion noise is high in actual aero-engines, but low in most laboratory combustion chambers because most chambers in laboratories operate at atmospheric pressure without exhaust nozzle, and are not followed by a mean flow acceleration. One exception is the experiment of Bake et al. [START_REF] Bake | The entropy wave generator (ewg): a reference case on entropy noise[END_REF] that intended to specifically focus on indirect combustion noise. This experiment, called the Entropy Wave Generator (EWG), was used by Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF] to validate the analytical tools created by Marble and Candel [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] in the choked supersonic case [START_REF] Leyko | Numerical and analytical investigation of the indirect combustion noise in a nozzle[END_REF][START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF], showing that indirect combustion noise can be correctly predicted using the analytical method. The present work focuses on the subsonic cases to explain the mechanisms generating noise in the experience and to analyse the compact nozzle hypothesis in this case. In Section II the experimental configuration of the EWG experiment is described, as well as an analysis of the boundary conditions and a consideration about the heating device. A numerical simulation of the complete set-up is performed in Section III A, the analytical method of Marble and Candel [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] is briefly outlined and applied to the experimental subsonic data in Section III B. The variant proposed by Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF] is also outlined. A study of the limits of the compact nozzle hypothesis is presented in Section III C and a semi-analytical approach using the linearized Euler equations through the nozzle is then proposed.

Results obtained with all these methods are presented in Section IV and compared to experiments performed by Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF]. Finally conclusions are presented in Section V.

II. The EWG experiment

The experiment performed by Bake et al. [START_REF] Bake | The entropy wave generator (ewg): a reference case on entropy noise[END_REF] at the DLR consists of a convergent-divergent nozzle with an electrical heating device placed at the inlet as sketched in Fig. 2 (geometrical parameters are summarized in Table 1). Depending on the mass flow rate the operating conditions can be varied from unchoked to choked flow with various exit Mach numbers, including supersonic flow. Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF] and Mühlbauer et al. [START_REF] Muhlbauer | Numerical investigation of the fundamental mechanism for entropy noise generation in aero-engines[END_REF] studied the supersonic case (with a normal shock after the choked nozzle). The present study is therefore restricted to the unchoked subsonic configuration, using mainly Reference Test Case 2 of [START_REF] Bake | The entropy wave generator (ewg): a reference case on entropy noise[END_REF], summarized in Table 2 . The mean steady flow is perturbed with a temperature pulse (Fig. 3) generated electrically by the heating device (located 100 mm upstream the nozzle throat) with a period of 1 second and a duration of 0.1 s. The acoustic waves generated are measured using four microphones placed at different sections at the outlet, shown in Fig. 2. For the present study, the microphone placed at a distance of 1150 mm downstream from the nozzle throat will be used to compare analytical and numerical results with the experiment.

III. Simulations of the EWG experiment

To understand the mechanisms of noise generation analytical and numerical tools are used to reproduce the pressure signal captured by the microphone placed at the outlet of the experimental device. First a full Euler simulation of the entire set-up is performed (Section III A). The analytical method of Marble and Candel [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF], based on the compact nozzle hypothesis, is then used to obtain a fully analytical solution of the problem (Section III B). This approach allows the separation of direct and indirect sources but is limited to low frequencies because of the compactness assumption.

To study the validity of this hypothesis, a code called SNozzle (a solver for the linearized Euler equations in the frequency domain) is finally used (Section III C). Table 3 summarizes the three methods used and delimits the considered domain in each case.

Method Type Domain Method A Numerical Simulation X = Xin to X = Xout Method B Analytical Method X = Xsc to X = Xout Method C quasi-1D linearized Euler equations X = Xsc to X = Xout
Table 3 Three methods used to study the EWG.

A. Numerical simulations

Numerical simulations are performed using a two-dimensional axisymmetric mesh for the unsteady compressible Euler equations expressed in the conservative form [START_REF] Schønfeld | Steady and unsteady flows simulations using the hybrid flow solver avbp[END_REF]. The singularity on the axis has been removed by considering a small cylinder on which a slip condition is applied. The final mesh of the actual EWG nozzle geometry, shown in Fig. 4 has approximately 10000 nodes with 1000 nodes in the axial direction and 10 in each section, which was shown to yield simulations independent of the grid size. 3D effects were shown to be negligible in this case [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF], and an axisymmetric mesh is enough to capture the noise generated by the heating device completely. The discretized geometry takes into account the complete nozzle, including the settling chamber (full domain plotted in Fig. 2, from X in to X out ). The maximum mesh size is 1 mm, enough to resolve the temperature pulse accurately. The Lax-Wendroff scheme is used to solve the Euler equations using the AVBP code [START_REF] Schønfeld | Steady and unsteady flows simulations using the hybrid flow solver avbp[END_REF]. In [START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF] this scheme was shown to handle the present temperature gra-dients correctly on the above fine grid. No significant differences have been seen with higher order

Taylor-Galerking schemes. No viscosity, turbulence or boundary layers are taken into account in order to isolate the noise generation of the heating device from other possible sources.

Fig. 4 Zoom on the nozzle mesh.

Reflection coefficients

Boundary conditions in the EWG experiment are not anechoic: Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF] and Mühlbauer et al. [START_REF] Muhlbauer | Numerical investigation of the fundamental mechanism for entropy noise generation in aero-engines[END_REF] show that reflecting conditions should be considered at the experiment outlet to mimic the set-up correctly. The reflecting properties of the experiment were measured at the experimental outlet [START_REF] Muhlbauer | Numerical investigation of the fundamental mechanism for entropy noise generation in aero-engines[END_REF] and it was found that they can be modelled using a first order filter, namely,

R out = -1 1 + 2ıω/κ . (1)
This boundary condition is imposed in the following analytical model at a distance l out of the nozzle. Tuning l out and κ is sufficient to reproduce the correct boundary condition [START_REF] Selle | The actual impedance of non-reflecting boundary conditions: implications for the computation of resonators[END_REF]. This was done by Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF] for the supersonic case, obtaining κ = 160 s -1 and l out = 2100 mm measured from the nozzle throat. The outlet reflection coefficients are plotted in Fig. 5 of [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF],

where the experimental data is compared with the first order filter obtaining a good agreement for the frequency range of interest.

In the computation, the boundary condition given by Eq. ( 1) is introduced using non-reflecting boundary conditions (Poinsot and Lele [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF]). Selle et al. [START_REF] Selle | The actual impedance of non-reflecting boundary conditions: implications for the computation of resonators[END_REF] showed that the above reflection coefficient can be retrieved in the simulation by imposing the incoming waves as a function of the difference between the local and the reference pressures, namely,

L -= 2κ∆t(p ref -p)/(ρc) , (2)
where ρ is the density and c the sound speed. When considering κ = 0 the boundary is completely non-reflecting, and when κ → ∞ waves are completely reflected. For any intermediate value, the boundary condition acts as the first order filter given in Eq. ( 1), [START_REF] Selle | Compressible large-eddy simulation of turbulent combustion in complex geometry on unstructured meshes[END_REF]. The above value of κ = 160 s -1 is used to mimic the experimental reflection coefficient. To impose the boundary condition at the correct distance l out from the nozzle throat, the computational domain is extended to the appropriate length (X out = l out ).

As the computational domain takes into account the settling chamber, the only inlet condition to impose corresponds to the inlet of the settling chamber (X = X in ) where a fully reflective inlet (R in = 1) is used.

Heating device

The heating device is computed as a source term in the energy equation. The energy is distributed spatially,

φ(x, t) = Φ 0 1 2 tanh x -x hd + l h /2 d tanh - x -x hd -l h /2 d + 1 φ(t) , (3)
where x hd = -100 mm is the heating device location, l h = 60 mm the heating device length, d = 4 mm a characteristic slope length and φ(t) the temporal fluctuation of the source term, given by an analytical function approximating the pulse generated, as done by Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF],

φ(t) =                1 -exp( t-t0 τ ) if t ∈ [t 0 , t 0 + T p ] φ(t 0 + T p ) exp(-t-t0 τ ) if t > t 0 + T p , (4)
where t 0 = 100 ms is the triggering time, T p = 100 ms the pulse duration and τ the relaxation time of the pulse, set to 7 ms in our case. The comparison of the experimental and the analytical temperature pulse is shown in Fig. 5.

The comparison of numerical simulations with the experimental data (Fig. 6) shows that the evolution of pressure at the microphone placed 1150 mm downstream the nozzle is reasonably captured. The peak values of the signal are slightly overpredicted (a similar accuracy is found in Fig. 6 of [START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF]). To obtain a better match in the shape, a more detailed model for the outlet reflection coefficient should be used instead of a first order filter modelization.

B. Analytical Method

The analytical approach aims at calculating the noise produced by the EWG using the analytical domain plotted in Fig. 2, from X sc to X out . The method is based on the derivation of jump conditions determining acoustic wave amplitudes at three locations: (0) and ( 1) upstream and downstream the heating device and (2) the outlet of the nozzle (at X out ) as shown in Fig. 2.

Heating device

Analytical methods are based on the decomposition of the source term in entropy and acoustic waves generated by the heating device. A simple 1D model can be used in the configuration illustrated in Fig. 7 to quantify the waves amplitudes: it takes into account the fluctuating heat release of the heating device Q , the acoustic and entropy waves (P + , P -and σ) and the mean flow. It should be noticed that the mean flow is constant through the device, and therefore M 0 = M 1 .

P + 0 P - 0 σ 0 P + 1 P - 1 σ 1
Fig. 7 Wave definition through the heating device.

As the heating device is short compared to all wavelength, the compact assumption used by Leyko et al. [START_REF] Leyko | Comparison of direct and indirect combustion noise mechanisms in a model combustor[END_REF] states that the wavelength of the perturbations is long compared to the heating device length, and therefore perturbations of mass, temperature and entropy are transmitted from one side to the other of the heating device without any time delay. Considering negligible mean heat release (Q 0), the conservation equations of mass enthalpy and entropy can be written between both sides of the heating device as a jump condition, namely,

ṁ ṁ 0 = ṁ ṁ 1 , (5)
T t T t 0 + q 1 + γ -1 2 M 2 0 -1 = T t T t 1 , (6)
s C p 0 + q = s C p 1 , (7)
where q = Q /( ṁ0 C p T 0 ) is a dimensionless form of the fluctuating heat release, s is the entropy fluctuation, C p is the specific heat capacity and γ the heat capacity ratio. Using isentropic relations, the ideal gas law, and writing the mass flow as ṁ = ρAu, the mass, total temperature and entropy equations can be expressed as a function of the primitive variables, namely,

ṁ ṁ = 1 M u c + p γp - s C p , (8)
T t T t = 1 1 + [(γ -1)/2]M 2 (γ -1)M u c + (γ -1) p γp + s C p , (9)
s C p = p γp - ρ ρ , (10)
where non-primed variables u, p, ρ and c represent the mean flow velocity, pressure, density and sound speed respectively, and primed variables represent small fluctuations of speed u , pressure p and density ρ .

Equations ( 5)-( 7) have to be written as a function of the waves shown in Fig. 7 to impose the boundary conditions properly. The amplitudes of these waves (P + , P -and σ) are written as a function of the primitive variables as,

P + ≡ 1 2 p γp + u c , (11)
P -≡ 1 2 p γp - u c , (12)
σ ≡ s C p = p γp - ρ ρ . (13)
corresponding to the upstream and downstream propagating acoustic waves and the entropy wave, which propagate at speeds u+c, u-c and u respectively (Fig. 7). Using Eqs.( 8)-(10) into Eqs.( 5)- [START_REF] Goldstein | A generalized acoustic analogy[END_REF], three relations between the known and unknown waves amplitudes are obtained,

(1 + 1 M 1 )P + 0 + (1 - 1 M 1 )P - 0 -σ 0 = (1 + 1 M 1 )P + 1 + (1 - 1 M 1 )P - 1 -σ 1 , (14)
(1 + M 1 )P + 0 + (1 -M 1 )P - 0 + σ 0 + q γ -1 = (1 + M 1 )P + 1 + (1 -M 1 )P - 1 + σ 1 γ -1 , (15)
σ 0 + q = σ 1 . (16)
To obtain the waves generated by the heating device the incoming waves (P + 0 , σ 0 , P - 1 ) are set to zero, so that the out-coming ones are caused by the effect of the heating device only (σ 1 = σ h ,

P + 1 = P + h , P - 0 = P - h ).
They read:

σ h = q , (17)
P + h = M 1 2(M 1 + 1) q , (18)
P - h = M 1 2(1 -M 1 ) q , ( 19
)
where the subscript h denotes the wave generated by the effect of q directly. Eqs. ( 17)- [START_REF] Lamarque | Boundary conditions for acoustic eigenmodes computation in gas turbine combustion chambers[END_REF] show that the heating device generates stronger entropy waves than acoustic ones when considering low Mach numbers (by a ratio σ h /P + h of approximately 60 for the EWG case: M 1 = 0.033).

Yet the contribution of direct noise at the outlet cannot be neglected without first considering a propagation model.

Reflection coefficients

To correctly model the propagation of waves through the EWG, the reflection coefficients have to be studied. The upstream reflection coefficient at the nozzle throat section (X = X 0 ) is noted R 1 , and the downstream reflection coefficient is noted R 2 (Fig. 8). The outlet reflection coefficient (R 2 ) is obtained by shifting (by a distance L out ) the reflection coefficient at the outlet of the numerical simulation, R out of Eq. ( 1) to section X 0 . R 2 is written,

R 2 = P - 2 P + 2 = P - out exp -2πif Lout c2(1-M2) P + out exp +2πif Lout c2(M2+1) = R out exp -2πif L out c 2 (M 2 + 1) exp -2πif L out c 2 (1 -M 2 ) , (20)
where L out = 2100 mm is the same distance used for the numerical simulation. Similarly, the inlet reflection coefficient R 1 can be written as a function of the reflection coefficient at X sc , R sc , shifted by L in ,

R 1 = R sc exp(-2πif L in c 1 (M 1 + 1) ) exp(-2πif L in c 1 (1 -M 1 )
) ,

where L in = 250 mm is the distance between the settling chamber outlet (X sc ) and the nozzle throat (X 0 ). For both cases the effect of the Mach number is retained for the propagation, though

for low Mach numbers it can be neglected, obtaining the same equations as in [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF]. The reflection coefficient R sc must be modelled. In the supersonic case Leyko et al. [START_REF] Leyko | Numerical and analytical investigation of the indirect noise in a nozzle[END_REF] tried both R sc = 0 and R sc = -1, showing that it had little influence on the results. For the present subsonic cases however, the settling chamber must be modelled as it is supposed to have a stronger effect on the results. In fact, as the flow is subsonic, waves can propagate upstream from the nozzle divergent section, reflect on the inlet at X sc and propagate downstream, interacting again with the nozzle and contributing to the global noise. Unfortunately, no experimental data exist for this reflection coefficient. An analytical approximation is proposed here to take into account the settling chamber.

The conservation equations across the section change at X sc between the settling chamber (SC) and the inlet duct (0) before the heating device (Fig. 2) can be written,

S sc • u sc = S 0 • u 0 , (22)
p sc = p 0 , (23)
considering low Mach number both at the settling chamber and at the nozzle inlet. In this way, two relations can be written between the waves at the settling chamber and at the heating device tube, namely,

P + sc -P - sc = Γ(P + 0 -P - 0 ) , (24)
P + sc + P - sc = P + 0 + P - 0 , (25)
where Γ = S0 Ssc is the ratio between sections. R sc is defined as the relation between the downstream propagating wave and the upstream propagating one in section X sc . This gives,

R sc = P + 0 P - 0 = (Γ -1) + R in (Γ + 1) (Γ + 1) + R in (Γ -1) . ( 26
)
where R in is the reflection coefficient at the inlet of the settling chamber (X in ) phase-shifted to X sc using the length of the settling chamber as done in Eq. 21. At the inlet of the settling chamber a fully reflective mass flow inlet is used, as for the numerical simulation, giving,

R in = 1 -M in 1 + M in exp(-2πif L in c 1 (M 1 + 1) ) exp(-2πif L in c 1 (1 -M 1 )
) ,

where M in and M sc are the Mach numbers at the inlet and inside of the settling chamber respectively.

It can be seen that for small frequencies R in tends to (1 -M in )/(1 + M in ), and knowing that M in is small, we obtain R sc ≈ 1. Using the numerical simulations the reflection coefficient can be calculated, showing a good agreement with the model.

Propagation

The propagation of waves through the nozzle of the EWG can be computed analytically in the low frequency limit [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] by writing the jump conditions through the nozzle using the compact nozzle hypothesis. This assumption states that the characteristic wavelength of the waves propagating through the nozzle λ is long compared to the nozzle length. In other words, the Helmholtz number (or dimensionless frequency),

Ω = L n λ = f L n c 1 , (28)
is negligible. L n = 263 mm is the nozzle length, c 1 is the sound speed at the inlet section of the nozzle (see Fig. 2) and f the frequency. For the EWG experiment, Fig. 3 shows that most of the spectral energy of the temperature pulse is contained at frequencies lower than 100 Hz, which gives a maximum Helmholtz number of Ω ≈ 0.077, so that the compact nozzle assumption is apparently reasonable. As the wavelength of the perturbations is long compared to the nozzle length, it can be assumed that the perturbations are transmitted from the inlet to the outlet with no time delay. The conservation equations of mass, enthalpy and entropy can be written as a simple jump condition for the fluctuations between the inlet and the outlet of the nozzle as done for the heating device,

ṁ ṁ 1 = ṁ ṁ 2 , (29)
T t T t 1 = T t T t 2 , (30)
s C p 1 = s C p 2 . ( 31
)
These jump conditions are written as a function of the waves through the nozzle (Fig. 8), using

Eqs. ( 8)-( 13), to impose the boundary conditions of the nozzle correctly. This yields,

(1 + 1 M 1 )P + 1 + (1 - 1 M 1 )P - 1 = (1 + 1 M 2 )P + 2 + (1 - 1 M 2 )P - 2 , (32)
(1 + M 1 )P + 1 + (1 -M 1 )P - 1 + σ 1 γ -1 1 + γ -1 2 M 2 1 = (1 + M 2 )P + 2 + (1 -M 2 )P - 2 + σ 2 γ -1 1 + γ -1 2 M 2 2 , (33)
σ 1 = σ 2 . (34)
For the subsonic case, three waves are incoming (P + 1 , σ 1 and P - 2 ) and have to be imposed, and three are out-coming, and therefore unknown (P - 1 , P + 2 and σ 2 ), as seen in Fig. 8. The problem can be solved using the three relations of Eq. ( 32)-(34), the incoming waves of Eqs. ( 17)-( 19) and the reflection coefficients R 1 and R 2 calculated in Section III B 2.

The downstream propagating acoustic wave at the inlet (P + 1 ) is calculated as the sum of the reflected wave and the total acoustic wave generated by the heating device at the inlet of the nozzle,

P + 1 = R 1 P - 1 + P h . (35)
The acoustic wave P h represents the direct noise source term, and is calculated as the combination of both acoustic waves generated by the heating device (P + h and P - h , using Eqs.( 18)-( 19))

using the inlet reflection coefficient. Knowing that q = σ h , it reads,

P h = P + h + R 1 • P - h = 1 2 M 1 M 1 + 1 + R 1 M 1 1 -M 1 σ h . (36)
A matrix system can be written to solve the problem considering both reflecting coefficients at the inlet and outlet of the nozzle R 1 and R 2 ,

            R 1 -1 0 0 0 0 -1 R 2 ξ - 1 ξ + 1 -ξ - 2 -ξ + 2 ζ 1 β - 1 ζ 1 β + 1 -ζ 2 β - 2 -ζ 2 β + 2                         P - 1 P + 1 P - 2 P + 2             =             0 0 0 ζ 2 -ζ 1             σ 1 -             0 0 ξ + 1 ζ 1 β + 1             P h , (37)
which can be reduced into a more compact form,

        ξ + 1 R 1 + ξ - 1 -(ξ + 2 + ξ - 2 R 2 ) ζ 1 (β + 1 R 1 + β - 1 ) -ζ 2 (β + 2 + β - 2 R 2 )                 P - 1 P + 2         =         0 ζ 2 -ζ 1         σ 1 -         ξ + 1 ζ 1 β + 1         P h , (38)
where ξ, β and ζ are a function of the Mach number only,

ξ ± = 1 ± 1 M , ( 39
)
β ± = (γ -1)(1 ± M ) , (40)
ζ = 1 + γ -1 2 M 2 -1 . ( 41
)
The problem is solved using the Fourier transform of the source terms and solving the matrix system to find P - 1 and P + 2 for each frequency. The pressure signal is obtained inverting Eqs. ( 11)-( 12) as,

p micro = (γp 2 ) P + micro + P - micro , (42)
and calculating the inverse Fourier transform. The subscript micro makes reference to the location of the microphone. The waves at the microphone location can be calculated with a simple phase-

shift, P + micro (x) = P + 2 exp[-2πif x/(c 2 (1 + M 2 ))] and P - micro (x) = P - 2 exp[-2πif x/(c 2 (1 -M 2 ))].
In Eq. ( 38), the two source terms σ 1 and P h are related to the indirect and the direct source terms respectively. To study the influence of indirect noise only, it is sufficient to set the other term to zero. An alternative method to solve the problem has also been proposed by Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF] considering no reflection coefficients and only indirect noise. This method is described in the appendix.

Figure 9 shows the results obtained using only the indirect source term (setting P h to zero).

Neither the shape nor the amplitude of the signal are correctly predicted. It should be noticed that the experimental noise level is 20 times larger than the predicted one and that the reflection coefficients should be taken into account to reproduce the experiment, as they have a strong influence on the signal. Figure 10 shows the solution of Eq. ( 38) with all source terms when using a non reflecting inlet and outlet boundary conditions (R 1 = 0, R 2 = 0) (Fig. 10(a)), a non-reflecting inlet and a partially reflective outlet (R 1 = 0, R 2 = 0) (Fig. 10(b)) and a reflecting inlet and outlet (R 1 = 0, R 2 = 0) (Fig. 10(c)).

The results show a good agreement when considering all reflection coefficients and both direct and indirect noise, though the peak values are over-predicted. The comparison of Fig. 9 and Fig. 10 shows that the influence of direct noise is stronger than the indirect noise. To illustrate the order of magnitude of both contributions, the indirect to direct noise ratio is considered and defined as in [START_REF] Leyko | Comparison of direct and indirect combustion noise mechanisms in a model combustor[END_REF], Indirect noise ratio

× σ h P + h Wave ratio × P + 2,dir P + 1 -1 Direct Noise ratio . ( 43
)
It represents the ratio between the acoustic waves generated at the outlet of the nozzle by indirect mechanisms, and those generated by direct ones. The 'Wave ratio' is the ratio between entropy and acoustic waves generated by the heating device (obtained by dividing Eqs. ( 17)-( 18)), giving,

σ h P + h = 2 1 + M 1 M 1 . (44)
The indirect (P + 2,ind /σ 1 ) and the direct (P + 2,dir /P + 1 ) noise ratio terms of Eq. ( 43) are the transfer functions of the nozzle. They are both calculated analytically using Eqs. (32)-(34): to calculate the indirect noise ratio, an entropy wave is considered while all other incoming waves are set to zero, and for the direct noise ratio only an incoming acoustic wave P + 1 is considered,

P + 2 σ 1 = M 2 -M 1 1 + M 2 1 2 M 2 1 + [(γ -1)/2]M 2 M 1 , (45)
P + 2 P + 1 = 2M 2 1 + M 2 1 + M 1 M 1 + M 2 1 + [(γ -1)/2]M 2 2 1 + [(γ -1)/2]M 2 M 1 . (46)
The noise ratio gives η ≈ 10 -2 , showing that indirect noise is negligible compared to direct noise, and that the measured pressure waves at the outlet are not caused by entropy waves but rather by the acoustics generated by the heating device.

C. SNozzle method

As stated before, the compact nozzle hypothesis is strictly valid only in the limit of infinite wavelength (zero frequency), though it has been used for small but non-zero frequencies. Leyko et al. [START_REF] Leyko | Comparison of direct and indirect combustion noise mechanisms in a model combustor[END_REF] showed that this assumption gives correct results for dimensionless frequencies up to Ω = 0.2 in the case of supersonic nozzles and subsonic convergent nozzles, but the case of a subsonic convergent divergent nozzle has never been considered. Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF] suggest that the compact nozzle hypothesis is no longer valid when considering a convergent-divergent flow and therefore that the results shown in Section III B might be misleading. Indeed the analytical equations obtained with the compact nozzle hypothesis (Eqs. ( 32)-( 34)) only take into account inlet to outlet relations, regardless of the flow evolution through the nozzle. In the reference case studied in this paper, the inlet and outlet Mach numbers are low, but the nozzle Mach number is large and the strong acceleration/deceleration produced inside the nozzle is not taken into account by the analytical model.

To study the validity of the compact assumption for non-zero frequencies in this type of nozzles, a numerical code based on the developments of Lamarque et al. [START_REF] Lamarque | Boundary conditions for acoustic eigenmodes computation in gas turbine combustion chambers[END_REF] called SNozzle will be used. The code solves the linearized Euler equations in the frequency domain, imposing the incoming waves, and using non-reflecting boundary conditions. The linearized Euler Equations (LEE) are written as,

∂ ∂t + u ∂ ∂x p γp + u ∂ ∂x u u = 0 , (47)
∂ ∂t + u ∂ ∂x u u + c 2 u ∂ ∂x p γp + 2 u u -(γ -1) p γp du dx = du dx s C p , (48)
∂ ∂t + u ∂ ∂x s C p = 0 , (49)
The indirect noise is caused by the last term of Eq. ( 48), where the entropy wave appears as a source term, which is non-zero when a mean flow gradient exists. The amplitude of the indirect noise directly depends on the mean flow acceleration, which is not seen by the analytical model, as it considers only a jump condition between the inlet and the outlet. The time derivatives are replaced by 2πif as the equations are solved in the frequency domain. Spatial derivatives are discretized with a central differencing scheme on a staggered 1D grid to prevent checkerboard oscillations.

The code is used to solve for the direct (P + 2,dir /P + 1 ) and indirect (P + 2,ind /σ 1 ) noise ratios of Eqs. ( 45)-( 46) in the SNozzle domain of Fig. 2 without the compact nozzle hypothesis. Results are plotted in Fig. 11 as a function of the reduced frequency for the EWG nozzle geometry. The compact nozzle solution (shown as a symbol) is only correct for the limit of very small Helmholtz numbers as the indirect noise ratio increases fast with frequency. From Ω ≈ 0.03 to the maximum dimensionless frequency of the signal (Ω ≈ 0.077), the indirect noise ratio is 20 times larger than the one predicted using the analytical method. The following physical interpretation is proposed: when considering zero-frequency, the entropy wave entering the nozzle generates a strong acoustic wave in the convergent, and another one in the divergent. These two waves have opposite phases due to the mean flow acceleration term in Eq. (48) (which is positive in the convergent, and negative in the divergent), and they cancel out at the outlet when they are added up, giving a small global contribution. When a non-zero frequency wave is considered, the phase-shift of both waves is not exactly π as the propagation of the waves through the nozzle has to be taken into account. Due to this change of phase, the two strong acoustic waves do not cancel out completely, generating an extra contribution due to the convergent-divergent effect. This extra contribution cannot be modelled by the compact nozzle method, as it takes into account only the global acceleration between the inlet and the outlet. This effect does not occur when the flow is only accelerated (or only decelerated).

The indirect to direct noise ratio calculated using SNozzle transfer functions is shown in Fig. 12.

For the frequency range of interest (Ω < 0.077), this ratio increases significantly, but the indirect noise is still smaller than direct noise, and therefore the direct noise generation cannot be neglected as already shown by the analytical method in Section III B. The large differences observed between the compact and the non-compact solutions at small reduced frequencies in Fig. 11 suggest that the compact nozzle hypothesis can be a source of error for convergent-divergent subsonic nozzles as the EWG. The code SNozzle has then been used to develop a semi-analytical method to calculate the pressure signal at the exit of the nozzle without the compact nozzle hypothesis. The propagation equations of the analytical method will be replaced

            R 1 -1 0 0 0 0 -1 R 2 0 P + 2 P + 1 P + 2 P - 2 -1 -1 P - 1 P + 1 P - 1 P - 2 0                         P - 1 P + 1 P - 2 P + 2             =             0 0 - P + 2 σ1 - P - 1 σ1             σ h +             0 0 - P + 2 P + 1 - P - 1 P + 1             P h . (55)
This system of equations is similar to the analytical one given above. Eq. ( 37) can be recovered if the numerical transfer functions are replaced by the analytical compact formula. The matrix and the source terms are all frequency-dependent, therefore the system has to be solved using the Fourier transform of the signal. The pressure signal at the microphone is again obtained with Eq. ( 42).

The experimental data is compared to the semi-analytical method in Fig. 13: no significant differences are observed. This suggests that the direct noise generated by the EWG experiment can be considered compact. Like for the numerical results of Fig. 6, the differences that can be still observed between the semi-analytical method and the experimental results are probably due to the modelling of the boundary conditions, especially the inlet reflection coefficient (R 1 ), where no experimental data is available.

V. Conclusions

The subsonic case of the EWG experiment performed at DLR has been studied using three Mach numbers the pulse generated by the heating device may shift towards higher frequencies as suggested by Howe [START_REF] Howe | Indirect combustion noise[END_REF], where non-compact effects are larger and tend to reduce the direct noise as seen in Fig. 11b. Using a first order analysis of the waves generated by the heating device it has been shown that, for the subsonic cases, direct noise is significant in the experimental set-up and that the indirect noise is negligible compared to it. The level of direct noise is even emphasized by the inlet reflection. These results differ from the choked case of [START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF] where indirect noise dominated direct noise.

indirect noise generation in the divergent.

This method provides the results shown in Fig. 14. It should be noticed that, for this analysis, direct combustion noise and reflecting boundaries were not taken into account. As stated before, the upstream propagating acoustic wave inside the nozzle has been neglected. It can be shown that this wave is significant and has to be included in the analysis. This wave, generated by entropy and acoustic waves propagating through the divergent, generates acoustic waves in the convergent section that propagate downstream contributing to the total noise. To solve the problem, Eq. ( 29)-( 31) can be written between the inlet (1) and the nozzle throat (T), and from the nozzle throat to the outlet [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF]. We obtain,

ṁ ṁ 1 = ṁ ṁ T = ṁ ṁ 2 , (60)
T t T t 1 = T t T t T = T t T t 2 , (61)
s C p 1 = s C p T = s C p 2 . ( 62
)
If the upstream propagating acoustic wave is not neglected, the method gives the same solution as the analytical solution of Marble and Candel. This occurs because the method proposed is still based on the compact nozzle hypothesis (even if it is applied to two separate elements, the method considers that the distance between them is zero, and the compact nozzle hypothesis could be applied to the whole system). A different solution would be obtained if the distance between the convergent and the divergent is considered different to zero. This last method was tested in the frame of this work, but was found to be too dependent on the considered length.

  Temporal variation of the electrical device source term φ(x, t) Source term in the energy equation Φ 0 Amplitude of the source term in the energy equation ρ Fluid mean density σ Entropy wave τ Relaxation time of the heating device pulse model I. Introduction

Fig. 1

 1 Fig. 1 The two mechanisms of combustion noise generation: Direct noise and indirect noise.

Fig. 2

 2 Fig.2Sketch of the EWG considered for the numerical simulations, with the definition of the analytical and the SNozzle domains, the sections where the different variables will be calculated: SC (settling chamber section at Xsc), 0 (inlet duct section at Xsc), 1 (inlet nozzle section) and 2 (exit nozzle section), and the four microphones at the exit.

Fig. 3

 3 Fig. 3 Experimental temperature pulse induced by the heating device. Left: time signal. Right: Fourier transform of the time signal.

Fig. 5

 5 Fig. 5 Experimental temperature pulse induced by the heating device. Analytical function modelling the temperature signal compared to the EWG experimental data.

Fig. 6

 6 Fig. 6 Numerical simulation pressure signal compared to experimental data from the EWG.

Fig. 8

 8 Fig. 8 Acoustic and entropy waves at the nozzle inlet and outlet in a subsonic nozzle. Waves are reflected in the boundary conditions Rsc and Rout.

Fig. 9

 9 Fig. 9 Experimental pressure signal obtained at the outlet microphone (Xmic = 1150 mm) compared to the analytical results considering indirect noise generation only (σ1, while P h = 0). Solid line: Experimental data (left scale), Dashed line: Analytical method (right scale).

  Fig. 10 Experimental pressure signal obtained at the outlet microphone (Xmic = 1150 mm) compared to the analytical results considering both direct (P h ) and indirect (σ1) noise. Solid line: Experimental data, Dashed line: Analytical method.

Fig. 11

 11 Fig. 11 Nozzle transfer functions calculated using SNozzle. Symbol: Analytical solution (Ω = 0). Solid line: SNozzle code solving Eqs. 47-49.

Fig. 12

 12 Fig. 12 Indirect to direct noise ratio η (Eq 43). Dot: Analytical solution (Ω = 0). Solid line: SNozzle solution solving the LEE (Eqs. 47-49).

Fig. 13

 13 Fig. 13 Pressure signal measured at the outlet for Reference Test Case 2 (Mach 0.7 at the throat). Solid line: Experimental data. Grey dashed line: Numerical simulations. Dark dashed line: Marble and Candel Method. Red dashed line: SNozzle method.

Fig. 14

 14 Fig. 14 Evolution of the noise peak at the outlet of the EWG as a function of the throat Mach number.

  different methods: numerical simulations of the Euler equations, analytical solution with Marble and Candel's model and the quasi one-dimensional linearized Euler equations model in the frequency domain. The direct noise has been shown to have an important effect on the global noise generated: the noise measured at the outlet comes from both direct and indirect noise generation, and is strongly modified by the reflection at the boundaries of the experimental set-up. The fully analytical model, based on the compact nozzle hypothesis, correctly predicts the shape of the pressure signal at the outlet when all reflection coefficients are included, but overpredicts the peak amplitude. The LEE code (SNozzle) shows that the limits of the analytical model to predict the indirect and direct noise generation for non-zero frequencies in this configuration may be caused by the compactness assumption in Marble and Candel's model which is no longer valid in the subsonic case for large frequencies. Successive stages of subsonic acceleration and deceleration of the mean flow are present in axial turbo-machines, and therefore a detailed study of these cases should be made to analyse the possible errors made when using the compact nozzle hypothesis to calculate the noise at the outlet of aero-engines. A semi-analytical method where the LEE solver is used to compute the transfer functions of the nozzle has been used to solve the propagation through the nozzle without the compact hypothesis with a strict separation of direct and indirect noise and taking into account the inlet and outlet reflection coefficients. This method gives the right trend for the reference test case and over a wide range of throat Mach numbers for the first time. It also shows that the direct noise generated in the EWG experiment can be considered compact and can be calculated using Marbel and Candel's analytical theory at low Mach numbers. However at larger

Table 1

 1 Geometric characteristics of the nozzle of the experimental set-up.

	Convergent Divergent Inlet	Outlet	Throat
	Length	Length Diameter Diameter Diameter
	13mm	250mm	30mm	40mm	7.5mm
	Inlet	Throat	Outlet	Inlet	Outlet
	Mach number Mach Number Mach number Pressure Pressure
	0.033	0.7	0.01861	105,640Pa 101,300Pa

Table 2

 2 Physical parameters of the subsonic case (Reference Test Case 2).

Acknowledgements

The authors would like to thank Nancy Kings and Friedrich Bake for providing the test case data and fruitful inputs. This work was partly funded by Snecma.

by the transfer functions of the nozzle, calculated using the 1D LEE code SNozzle in the restricted 'SNozzle domain' plotted in Fig. 2. This will be combined with the same reflection coefficients at the inlet and the outlet as in Section III B and the same analytical model for the heating device.

In this way the non-compactness of the nozzle alone can be assessed accurately. To yield the complete semi-analytical model, the transfer functions, defined as the out-coming waves generated by a unitary incoming wave, are calculated using non reflecting boundary conditions at all boundaries.

Combining the incoming and out-coming waves, nine transfer functions are then obtained, namely,

for the upstream propagating acoustic wave,

for the entropy wave, and,

for the downstream propagating acoustic wave. These transfer functions are frequency-dependent.

To calculate them, three sets of simulations were performed using SNozzle. In each one, a single unitary wave (entropy σ 1 , acoustic at the inlet P + 1 or acoustic at the outlet P - 2 ) is introduced to calculate the out-coming ones. For each set of simulations, a frequency sampling was done. In this way each SNozzle simulation has a single unitary incoming wave at a single frequency and the transfer functions of Eqs. ( 50)-( 52) can be calculated independently as a function of frequency. The results obtained are used to solve the EWG problem including the waves and reflection coefficients.

The linear property of the LEE solution is first used to write,

Writing the acoustic waves at the inlet and at the outlet as

two extra equations are obtained. The system can be re-arranged in a matrix form as,

To study the subsonic convergent divergent nozzle, a method was proposed by Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF] to solve the propagation analytically using the compact nozzle hypothesis without the limitations of Marble and Candel's theory. This method is based on the division of the nozzle into two subsonic nozzles, one convergent and another one divergent. To solve the propagation of the waves, the quasi-transfer of the nozzle is calculated separately for the convergent and for the divergent and the propagation is studied sequentially. Yet this combination of transfer functions is actually coupled: the noise generated in the convergent depends on the reflection of the waves in the divergent, and vice versa. To solve the system of equations, Bake et al. [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF] thus neglected the upstream propagating acoustic wave (P - 1 ) generated in the divergent (by entropy and acoustic waves).

In this way the system of equations is no longer coupled. The quasi-transfer functions obtained can be written in the non-dimensional form using Marble and Candel's method (Eqs. ( 29)-( 31)) between the inlet (1) and the nozzle throat (T) for the convergent, and between the throat and the outlet

(2) for the divergent. In the case of the convergent they read,

and similarly for the divergent, replacing the inlet and throat subscripts for the throat and the outlet respectively. In dimensional form Eqs. ( 56)-(58) are equivalent to Eqs. ( 1)-( 4) of [START_REF] Bake | Experimental investigation of the entropy noise mechanism in aero-engines[END_REF]. The pressure wave at the outlet generated by the entropy wave is calculated combining the transfer functions,

The first term is the acoustic wave generated in the convergent which is obtained by the combination of two terms: the indirect noise generation in the convergent, and the propagation of this noise through the divergent. The second term is the indirect noise generated in the divergent, which is obtained combining the propagation of the entropy wave through the convergent with the