宇宙航空研究開発機構特別資料JAXA Special Publication

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宇宙航空研究開発機構特別資料

JAXA Special Publication

2011 年 3 月

宇宙航空研究開発機構

Japan Aerospace Exploration Agency

宇宙航空研究開発機構特別資料

東京大学　ロケットエンジンモデリングラボラトリー

（JAXA社会連携講座）　シンポジウム

ロケットエンジン解析技術の新展開

" New Horizon of Rocket Engine Modeling and Simulation "

後刷集

東京大学工学系研究科

　　ロケットエンジンモデリングラボラトリー（JAXA 社会連携講座）

宇宙航空研究開発機構　情報・計算工学（JEDI）センター　

JAXA-SP-10-010

This document is provided by JAXA.

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࡚ࠪࡦߩޟᣂዷ㐿ޠࠍ࿑ࠆߚ߼ߩ߈ߞ߆ߌࠍߟ߆߻ߎߣࠍ⋡⊛ߣߒߡડ↹ߐࠇޔ᧲੩ᄢቇ Ꮏቇ♽⎇ⓥ⑼ࡠࠤ࠶࠻ࠛࡦࠫࡦࡕ࠺࡝ࡦࠣ࡜ࡏ࡜࠻࡝࡯㧔 JAXA ␠ળㅪ៤⻠ᐳ㧕ߣቝቮ⥶ⓨ

⎇ⓥ㐿⊒ᯏ᭴㧔 JAXA 㧕ߩᖱႎ࡮⸘▚Ꮏቇ㧔 JEDI 㧕࠮ࡦ࠲࡯ߩ౒௅ߢ㐿௅ߐࠇ߹ߒߚޕ JAXA ␠ળㅪ៤⻠ᐳߪޔࡠࠤ࠶࠻ࠛࡦࠫࡦࠪࡒࡘ࡟࡯࡚ࠪࡦᛛⴚߩ㜞ᐲൻߦ㑐ߔࠆᢎ⢒࡮

⎇ⓥࠍኻ⽎ߣߒߡ㨬 10 ᐕ㑆ߢ਎⇇࠻࠶ࡊࠢ࡜ࠬߩᛛⴚജࠍขᓧߔࠆ㨭ߎߣࠍ⋡ᮡߣߒߡޔ 2008 ᐕ 4 ᦬ࠃࠅᵴേࠍᆎ߼ޔᧄᐕߩ 9 ᦬ߢߜࠂ߁ߤ 2 ᐕඨ߇⚻ㆊߒ߹ߒߚޕᧄ⻠ᐳߪᭂૐ

᷷߿㜞࿶ߥߤߩᶧ૕ࡠࠤ࠶࠻ࠛࡦࠫࡦ․᦭ߩⅣႺਅߦ߅ߌࠆ‛ℂ࡮ൻቇ⃻⽎ߩ⸃᣿ޔ߅ࠃ

߮ߘࠇߦၮߠ޿ߚᢙቇࡕ࠺࡞ߩ᭴▽ࠍ⋡ᜰߒߡ޿߹ߔޕ੹ޔ⻠ᐳߦߪ(1)ࠠࡖࡆ࠹࡯࡚ࠪࡦޔ (2) ྃ኿࡮ᓸ☸ൻޔ (3) 㜞࿶Ά὾ޔ (4) ‛ᕈ࡮ࡒࠢࡠ⃻⽎ߩ 4 ߟߩ⎇ⓥࠣ࡞࡯ࡊ߇޽ࠅ߹ߔޕߎ ࠇࠄߩ⎇ⓥࠣ࡞࡯ࡊߪᧄ⻠ᐳߣ JEDI ߣ߇ਥ૕ߦߥࠅޔ᧲ᄢߩ⥶ⓨቝቮᎿቇኾ᡹ޔᯏ᪾Ꮏቇ ኾ᡹ࠍߪߓ߼ߣߒߡޔઁᄢቇ࡮⎇ⓥᯏ㑐߅ࠃ߮㑐ㅪડᬺߩ⎇ⓥ⠪ߦࠃࠅ᭴ᚑߐࠇߡ޿ߡޔ ቯᦼ⊛ߦ㐿௅ߐࠇࠆ⎇ⓥળࠍਛᔃߣߒߡᵴേߒߡ߅ࠅ߹ߔޕᧄࠪࡦࡐࠫ࠙ࡓߪᧄ⻠ᐳߩᦨ

ೋߩ 2 ᐕඨߦ߅ߌࠆߎ߁ߒߚᵴേߩਛ㑆ႎ๔ߢ߽޽ࠅ߹ߔޕ

߹ߚޔᧄ⻠ᐳߪ᧲੩ᄢቇߩޟ␠ળㅪ៤⻠ᐳޠߩ৻ߟߢ޽ࠅޔ␠ળㅪ៤⻠ᐳߪᧄቇߦߢ߈ ߚޔ⎇ⓥߣᢎ⢒ߩᯏ⢻ࠍ૬ߖᜬߟᣂߒ޿઀⚵ߺߩ⻠ᐳߢߔޕᄢቇߦ߅ߌࠆၮ␆⎇ⓥߩࡐ࠹

ࡦࠪࡖ࡞ࠍ↢߆ߒޔ಴ญࠍ⷗ᝪ߃ߚ⎇ⓥᵴേࠍⴕ߁ߎߣ߇␠ળㅪ៤⻠ᐳߩ૶๮ߢ޽ࠆߣᕁ

޿߹ߔޕᧄ JAXA ␠ળㅪ៤⻠ᐳߦ߅ߌࠆ⎇ⓥߩ⋡⊛ߪ᣿⏕ߦቯ⟵ߢ߈ߡ߅ࠅޔ߹ߚߘߩߚ

߼ߩၮ␆⎇ⓥߩࡐ࠹ࡦࠪࡖ࡞߽᦭ߒߡ߅ࠅ߹ߔޕߒ߆ߒߥ߇ࠄޔ⋡ᮡࠍ㆐ᚑߔࠆߚ߼ߩ⎇

ⓥ૕೙ߪ޿߹ߛචಽߣߪ⸒߃ߥ޿ὐ߽޽ࠅޔᄙߊߩ⸃᳿ߔߴ߈⺖㗴ࠍᛴ߃ߡ޿ࠆߩ߽੐ታ ߢߔޕᧄࠪࡦࡐࠫ࠙ࡓߢ޿ߚߛ޿ߚߏᛕ್߿ߏᗧ⷗ߪޔߎࠇࠄߩ⺖㗴ߩ⸃᳿ߩ߈ߞ߆ߌߣ ߒߡ↢߆ߒߚ޿ߣᕁߞߡ߅ࠅ߹ߔޕ

ᧄࠪࡦࡐࠫ࠙ࡓߩ㐿௅ߦᒰߚࠅޔ⛘ᄢߥߏදജࠍ޿ߚߛ޿ߚ᜗ᓙ⻠Ṷޔၮ⺞⻠Ṷޔ৻⥸

⻠Ṷߩ⻠Ꮷߩ⊝᭽ޔߏෳട޿ߚߛ޿ߚ⊝᭽ޔߘߒߡ੐ോዪߩ⊝᭽ߦߎߩ႐ࠍ߅୫ࠅߒߡᷓ

ߊᓮ␞↳ߒ਄ߍ߹ߔޕ

ߥ߅ޔᧄ⾗ᢱߪ⻠Ꮷߩ⊝᭽ߩߏෘᗧߦࠃࠅޔ⊒⴫⾗ᢱࠍ pdf ࡈࠔࠗ࡞ߦᄌ឵ߒߡޔශ೚ߒ ߚ߽ߩߢߔ߇ޔᔅߕߒ߽ࠝ࡝ࠫ࠽࡞ߩ⊒⴫⾗ᢱࠍౣ⃻ߒ߈ࠇߡ޿ߥ޿ㇱಽ߇ᱷࠆߣߣ߽ߦޔ

േ↹ߦߟ޿ߡߪ⚕㕙ߢߪഀᗲߖߑࠆࠍᓧ߹ߖࠎߢߒߚޕߎࠇࠄߩὐߦߟ޿ߡߪޔߏኈ⿔ࠍ ߅㗿޿޿ߚߒ߹ߔޕ

᧲੩ᄢቇᎿቇ♽⎇ⓥ⑼ JAXA ␠ળㅪ៤⻠ᐳ

․છᢎ᝼ ⿧ శ↵

-i-

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ロケットエンジンモデリングラボラトリー（ＪＡＸＡ社会連携講座）

シンポジウム｢ロケットエンジン解析技術の新展開」

目 �

【基調講演】

(1)

キャビテーション流れのマルチスケール解析

1

東京大学工学系研究科松本洋一郎教授

(2)

ものづくり分野における最先端シミュレーションの展望

35

東京大学生産技術研究所加藤千幸教授

（革新的シミュレーション研究センター長）

(3)

東京大学航空宇宙工学専攻におけるロケット推進の研究活動

61

東京大学工学系研究科航空宇宙工学専攻渡辺紀徳教授

【招待講演】

(1) Overview of Pratt & Whitney Rocketdyne Modeling & Simulation

Practices for Liquid Propellant Rocket Engines

81 Dr. Munir M. Sindir, Pratt & Whitney Rocketdyne

(2)Astrium Space Transportation's Liquid Propulsion Heritage

and Simulation Capabilities

97 Dr. Oliver Knab, EADS Astrium

【一般講演】

(1) JAXA

の目指すロケットエンジンシミュレーション

115

宇宙航空研究開発機構情報・計算工学センター山西伸宏

(2)

ロケットエンジン開発における全系解析に向けた取り組み

125

宇宙航空研究開発機構情報・計算工学センター谷直樹

(3)

再生冷却エンジン燃焼室熱流束を支配する燃焼現象把握およびモデル評価

137

宇宙航空研究開発機構情報・計算工学センター大門優

(4) Overview of CRUNCH CFD Simulations For Liquid Propulsion Systems 147 Dr. Ashvin Hosangadi, CRAFT Tech

(5)

高解像度コンパクト差分法を用いた超臨界圧環境下における噴流構造の数値解析

163

宇宙航空研究開発機構情報・計算工学センター寺島洋史

(6)

燃焼振動抑制に向けた取り組み―レゾネータと燃焼器の減衰特性評価―

175

宇宙航空研究開発機構情報・計算工学センター清水太郎

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(7) 187

宇宙航空研究開発機構

宇宙輸送ミッション本部エンジン研究開発グループ冨田建夫

(8) LE-X

エンジンにおけるシミュレーション解析技術の活用

195

宇宙航空研究開発機構

宇宙輸送ミッション本部エンジン研究開発グループ野田慶一郎

(9) MHI

における液体ロケットエンジン設計へのシミュレーション適用の取組み

207

三菱重工業株式会社名古屋誘導推進システム製作所小金澤崇

(10)

ロケットエンジン設計解析技術－現在と未来－

219

株式会社

IHI

都丸裕司

(11)

水素の高圧燃焼反応機構

231

東京大学工学系研究科総合研究機構越光男教授

(12)

ロケット燃焼解析に必要な熱物性モデルと数値解析

239

東京大学工学系研究科総合研究機構清水和弥助教

(13)

推進薬の高圧噴射と微粒化に関連した熱流動現象

251

東京大学工学系研究科航空宇宙工学専攻姫野武洋准教授

(14)

多重プロセス型キャビテーションモデルの現状と課題

267

信州大学工学部津田伸一助教

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プログラム

【平成２２年９月２８日(火) 10:30-17:40】

10:30 - 10:40 開会挨拶北森武彦（東京大学大学院工学系研究科長）

JAXA の目指すロケットエンジンシミュレーション山西伸宏（宇宙航空研究開発機構）

ロケットエンジン開発における全系解析に向けた取り組み谷直樹（宇宙航空研究開発機構）

再生冷却エンジン燃焼室熱流束を支配する燃焼現象把握およびモデル評価大門優（宇宙航空研究開発機構）

10:40 - 12:00 一般講演 1

Overview of CRUNCH CFD Simulations For Liquid Propulsion Systems Ashvin Hosangadi (CRAFT Tech)

高解像度コンパクト差分法を用いた超臨界圧環境下における噴流構造の数値解析寺島洋史（宇宙航空研究開発機構）

燃焼振動抑制に向けた取り組み―レゾネータと燃焼器の減衰特性評価―

清水太郎（宇宙航空研究開発機構）

13:20 - 14:20 一般講演 2

ロケット燃焼器の燃焼試験技術の現状と課題冨田健夫（宇宙航空研究開発機構）

LE-X エンジンにおけるシミュレーション解析技術の活用野田慶一郎（宇宙航空研究開発機構）

MHI における液体ロケットエンジン設計へのシミュレーション適用の取組み小金澤崇（三菱重工業株式会社）

14:30 - 15:30 一般講演 3

ロケットエンジン設計解析技術－現在と未来－

都丸裕司（株式会社 IHI）

15:40 - 16:40 招待講演 1

Overview of Pratt & Whitney Rocketdyne Modeling & Simulation Practices for Liquid Propellant Rocket Engines

Dr. Munir M. Sindir (Pratt & Whitney Rocketdyne) 16:40 - 17:40

招待講演 2

Astrium Space Transportation's Liquid Propulsion Heritage and Simulation Capabilities Dr. Oliver Knab (EADS Astrium)

【平成２２年９月２９日(水) 9:00-14:50】

9:00 - 10:00 基調講演 1

キャビテーション流れのマルチスケール解析松本洋一郎教授（東京大学）

10:00 - 11:00 基調講演 2

ものづくり分野における最先端シミュレーションの展望加藤千幸教授（東京大学）

11:00 - 12:00 基調講演 3

東京大学航空宇宙工学専攻におけるロケット推進の研究活動渡辺紀徳教授（東京大学）

水素の高圧燃焼反応機構

越光男教授（東京大学）

ロケット燃焼解析に必要な熱物性モデルと数値解析清水和弥助教（東京大学）

推進薬の高圧噴射と微粒化に関連した熱流動現象姫野武洋准教授（東京大学）

13:20 - 14:40 一般講演 4

多重プロセス型キャビテーションモデルの現状と課題津田伸一助教（信州大学）

14:40 - 14:50 閉会挨拶嶋英志（宇宙航空研究開発機構情報・計算工学センター長）

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⊒ ⴫ ౝ ኈ

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䉨䊞䊎䊁䊷䉲䊢䊮ᵹ䉏䈱䊙䊦䉼䉴䉬䊷䊦⸃ᨆ

᧲੩ᄢቇᎿቇ♽⎇ⓥ⑼

᧻ᧄᵗ ㇢

᧻ᧄᵗ৻㇢

⋡ᰴ

• 䈲䈛䉄䈮

න ᳇ᵃ䈱᜼േ

• න৻᳇ᵃ䈱᜼േ

• ᳇ᵃ䉪䊤䉡䊄䈱᜼േ

• ᵹ૕ᯏ᪾䈱䉨䊞䊎䊁䊷䉲䊢䊮䉣䊨䊷䉳䊢䊮 ᵹ૕ᯏ᪾ 䉨䊞䊁

• 䈍䉒䉍䈮䈍䉒䉍䈮

㻝

(8)

䉨䊞䊎䊁䊷䉲䊢䊮䈮䈍䈔䉎䊙䊦䉼䉴䉬䊷䊦䉻䉟䊅䊚䉾䉪䉴

B-B Interaction Volumetric oscillation

Mist

Mist Temperature

Distribution

Internal Phenomena

of a Cavitation Bubble Bubble Cloud Cavitating Flow around a Hydrofoil

Micro Mezzo Macro

of a Cavitation Bubble around a Hydrofoil

Micro Mezzo Macro

⋡ᰴ

• 䈲䈛䉄䈮

න ᳇ᵃ䈱᜼േ

• න৻᳇ᵃ䈱᜼േ

• ᳇ᵃ䉪䊤䉡䊄䈱᜼േ

• ᵹ૕ᯏ᪾䈱䉨䊞䊎䊁䊷䉲䊢䊮䉣䊨䊷䉳䊢䊮 ᵹ૕ᯏ᪾ 䉨䊞䊁

• 䈍䉒䉍䈮䈍䉒䉍䈮

㻞

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Time history of bubble radius Time history of bubble radius

A

A E E

R

R 13 13

A

A B B D D R R ₀ ₀ = 13 = 13 ��m m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀ ₀ = 100 kPa = 100 kPa

��pp = 50 kPa = 50 kPa

C C

��pp 50 kPa 50 kPa

C C Mist

Mist

²⁴⁰²⁴⁰ ^{840 [K]}^{840 [K]}

A

A B B C C D D E E

Temperature distribution inside bubble Temperature distribution inside bubble

Assumptions Assumptions

(1)

(1) Gases inside the bubble and the surrounding liquid move Gases inside the bubble and the surrounding liquid move maintaining spherical symmetry

maintaining spherical symmetry maintaining spherical symmetry.

maintaining spherical symmetry.

(2)

(2) Gases inside the bubble obey the perfect gas law. Gases inside the bubble obey the perfect gas law.

(3)

(3) Non Non--condensable gas obeys Henry’s law at the bubble wall. condensable gas obeys Henry’s law at the bubble wall.

(4)

(4) Classical theory for generation and growth of mist under quasi Classical theory for generation and growth of mist under quasi-- equilibrium condition is applied, because the temperature inside equilibrium condition is applied, because the temperature inside the bubble does not change so rapidly .

the bubble does not change so rapidly . (5)

(5) Coalescence and fragmentation of the mist are ignored. Coalescence and fragmentation of the mist are ignored.

(6)

(6) Mist has the same velocity as the gas mixture and the effect of Mist has the same velocity as the gas mixture and the effect of diffusion by Brownian motion is assumed to be small and diffusion by Brownian motion is assumed to be small and ignored.

ignored.

(7)

(7) Viscosity of the liquid is ignored except at the bubble wall. Viscosity of the liquid is ignored except at the bubble wall.

㻟

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Governing equations of Governing equations of Direct Numerical Simulation (1) Direct Numerical Simulation (1) ( ) ( )

Using simulation code developed by Takemura & Matsumoto (1994) Using simulation code developed by Takemura & Matsumoto (1994)

䂾

Full conservation equations in gas phase with mist Full conservation equations in gas phase with mist

Using simulation code developed by Takemura & Matsumoto (1994) Using simulation code developed by Takemura & Matsumoto (1994)

䊶

Conservation equation of Mass Conservation equation of Mass

䊶

Conservation equation of Momentum Conservation equation of Momentum

䊶

Conservation equation of Energy Conservation equation of Energy Conservation equation of Energy Conservation equation of Energy

䂾

Nucleation rate equation of mist by homogeneous condensation Nucleation rate equation of mist by homogeneous condensation

䂾

Conservation equation of number density of mist Conservation equation of number density of mist

䂾

Energy equation in liquid phase Energy equation in liquid phase

䂾

Diffusion equation of non Diffusion equation of non condensable gas in liquid condensable gas in liquid

䂾

Diffusion equation of non Diffusion equation of non--condensable gas in liquid condensable gas in liquid

Governing equations of Governing equations of Direct Numerical Simulation (2) Direct Numerical Simulation (2) ( ) ( ) Motion of bubble wall

Motion of bubble wall

䂾

䂾 Equation of bubble wall motion Equation of bubble wall motion 䋨䋨 Fujikawa & Akamatsu, 1980 Fujikawa & Akamatsu, 1980 䋩䋩

�

� Considered Considered 䊶

䊶 Liquid compressibility (1st Liquid compressibility (1st order approximation) order approximation) 䊶

䊶 Phase change at bubble wall Phase change at bubble wall

㻠

(11)

Time history of bubble radius Time history of bubble radius

R

R = 13 = 13 ��m m R

R ₀ ₀ 13 13 ��m m ff _d _d = 0.2 MHz = 0.2 MHz

100 k 100 k pp ₀ ₀ = 100 kPa = 100 kPa

�� p p = 50 kPa = 50 kPa

㪩㪅㪄㪧㪅㩷㪜㫈㫅㪅䋭㪠㫊㫆㫋㪿㪼㫉㫄㪸㫃

㪪㫀㫋㪿㪻䋭㪪㫎㫀㫋㪺㪿㪼㪻䋭㪘㪻㫀㪸㪹㪸㫋㫀㪺

Bubble motion Bubble motion

Comparison between simulation and experiment Comparison between simulation and experiment

㻡

(12)

Maximum bubble radius Maximum bubble radius

R

R ₀ ₀ = 3 ~ 40 = 3 ~ 40 �� m m ff _dd = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀

��p p = 50 kPa = 50 kPa

Time histories of bubble radius, temperature Time histories of bubble radius, temperature

and mist density inside bubble and mist density inside bubble and mist density inside bubble and mist density inside bubble R

R ₀ ₀ = 13 = 13 ��m, m, ff _dd = 0.2 MHz, = 0.2 MHz, pp ₀ ₀ = 100 kPa, = 100 kPa, ��p p = 50 kPa = 50 kPa

㻢

(13)

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

A A A A R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

�� p p = 50 kPa = 50 kPa

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

B B B B R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

��p p = 50 kPa = 50 kPa

㻣

(14)

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

C C C C R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

�� p p = 50 kPa = 50 kPa

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

D D D D R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

��p p = 50 kPa = 50 kPa

㻤

(15)

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

E E E E R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

�� p p = 50 kPa = 50 kPa

Distributions of temperature, vapor Distributions of temperature, vapor

concentration and mist density concentration and mist densityyy

FFFF R

R ₀₀ ₀₀ = 13 = 13 ��m �� m ff _d _d = 0.2 MHz = 0.2 MHz pp ₀₀ = 100 kPa = 100 kPa pp ₀₀ 100 kPa 100 kPa

��p p = 50 kPa = 50 kPa

㻥

(16)

Maximum bubble radius Maximum bubble radius

R

R ₀ ₀ = 0.3 ~ 4 = 0.3 ~ 4 ��m m ff = 2 MHz = 2 MHz ff _dd = 2 MHz = 2 MHz pp ₀ ₀ = 100 kPa = 100 kPa

Spherical bubble model Spherical bubble model

Internal thermal phenomena are considered Internal thermal phenomena are considered

�

� Mass and Mass and heat transfer through the bubble wall heat transfer through the bubble wall

�

� Phase change at the bubble wall Phase change at the bubble wall

�

� Counter diffusion of vapor and non Counter diffusion of vapor and non condensable gas condensable gas

�

� Counter diffusion of vapor and non Counter diffusion of vapor and non--condensable gas condensable gas

�

� Mist condensation and evaporation Mist condensation and evaporation Matsumoto,

Matsumoto, Trans. of JSME Trans. of JSME, 1986. , 1986.

�

� Temperature gradient model at the bubble wall Temperature gradient model at the bubble wall Preston et al.,

Preston et al., CAV2003, CAV2003, 2003. 2003.

Mist

㻝㻜

(17)

Governing equations Governing equations

The motion of the bubble wall (Fujikawa & Akamatsu equation)

3 4 3 1 4 2 2 3

1 ²

� �

�

��

�

� � �

��

�

��

�

� � �

p R p m p

m m R R m

c R c R m

c m c R R R

R l R l l

l �

�

� �

�

� �

�

2 0 2

1

�� ^, � ^, �

��

�

� �

��

�

��

�

� � �

� ^� ^� ^�

c p R p m p

m R c m c R R m

l R r l l

R r l l

l l

l � � � � �

� ^�

� �

�

^� ^� ^��_�^� ^� ^��_�^�

�

� �

�

� �

l l b i i l

l gi vi g v R r l

R m R R p m

p

p �

�

� � �

� 2 4

2

, �_l

�

�_vi��_gi

�

R_b R_� �_l_�

The energy conservation equation in gas phase with mist

� �

0

2 2

�

��

� �

�

� �

�

h dM dM h dM

h dM S T

LdM

dt d M p dt d M p dt

T M d C M C M C

c l c g v

c

v v

v g v g

g g c vl v vv g vg

�

� 0

��

� �

�

� _� � h dt

dt h dt

h dt S r

L dt _g _v _l

R r

�

The energy conservation equation in liquid phase The diffusion equation of non condensable gas in liquid

The diffusion equation of non-condensable gas in liquid The nucleation rate equation of mist

Present model vs. DNS Present model vs. DNS

Internal gas: nitrogen Internal gas: nitrogen Initial bubble radius: 2 Initial bubble radius: 2 ��m m Initial pressure:

Initial pressure: 100 kPa 100 kPa I iti l t t 293 K I iti l t t 293 K Initial temperature: 293 K Initial temperature: 293 K

DNS: Takemura and Matsumoto, DNS: Takemura and Matsumoto, JSME I J

JSME I J 1994 1994 JSME Int. J.

JSME Int. J.,, 1994. 1994. 100 kPa � 90 kPa � 100 kPa 100 kPa � 90 kPa � 100 kPa

100 kPa � 50 kPa � 100 kPa

100 kPa � 50 kPa � 100 kPa 100 kPa � 10 kPa � 100 kPa 100 kPa � 10 kPa � 100 kPa

㻝㻝

(18)

Heating Mechanism of Microbubbles Heating Mechanism of Microbubbles

Energy affecting the heat near the bubble

4 R

2

W � ^� T

^l

� Heat deposition

4 R W

_T

�

_l

r

^l

� �

�

Viscous dissipation 4

2

4 R R

R

W

_V

� �

_l

R � � � � � Viscous dissipation R

R

Acoustic emission

�

l

T

l

: thermal conduction : temperature

� �

2

2 2

2 4 c R

R R

W

_A

� �

_l

R � � � � � � Acoustic emission

T

l

�

l

�

l

: temperature : viscosity

: density c

_l

Energy escaping to large distances

c

l

: sound speed

Energy Radiation Energy Radiation

Frequency : 1.0 MHz Amplitude : 100 kPa Initial radius : 1 10 �m Initial radius : 1 - 10 �m Internal gas : air

㻝㻞

(19)

Properties of Microbubbles Properties of Microbubbles

Bubble radius

Ტ᳸ 10�m Უ

Shell

Ტ � Უ

Ტ material, thickness … Უ

Internal gas microbubble

Specific

heat ratio Gas

constant Heat conductivity heat ratio

[-] constant

[J / kg 䊶 K] conductivity [ ^{mW /m}

^䊶

^K ]

Argon (Ar) 1 67 208 1 18 2

Argon (Ar) 1.67 208.1 18.2

air 1.40 287.0 26.9

Sulfur Hexafluoride (SF ⁶ ) 1 09 56 9 14 8 Sulfur Hexafluoride (SF ⁶ ) 1.09 56.9 14.8

Ar bubble Ar bubble

Frequency : 1.0 MHz A lit d 100 kP Amplitude : 100 kPa Initial radius : 1 - 10 �m Internal gas : Ar (Air bubble)

(Air bubble)

e a gas :

㻝㻟

(20)

SF

SF 66 Bubble Bubble

Frequency : 1.0 MHz A lit d 100 kP Amplitude : 100 kPa Initial radius : 1 - 10 �m Internal gas : Ar (Air bubble)

(Air bubble)

e a gas :

Acoustic emission from a microbubble Acoustic emission from a microbubble

Acoustic velocity of water

=1.478 㫏 10

³

(m/s)

Emitted acoustic pressure from micro bubble in far field 䋨Fujikawa, 1979䋩

䋨Fujikawa, 1979䋩

㻝㻠

(21)

Nonlinear oscillation of a microbubble Nonlinear oscillation of a microbubble

R

₀₀

= = 1.5 1.5 ��m, m, pp

₀₀

= = 101.3 101.3 kPa, kPa, ff

₀₀

= = 22 MHz, MHz, ��p p = = 100 100 kPa kPa

mbient sure [-]am press

e [-]

bubble radius [

oustic uaco re [Pa]pressu

Time history of ambient pressure, bubble radius Time history of ambient pressure, bubble radius and acoustic pressure from the bubble

and acoustic pressure from the bubble

Nonlinear oscillation of a microbubble Nonlinear oscillation of a microbubble

R

₀₀

= = 1.5 1.5 ��m, m, pp

₀₀

= = 101.3 101.3 kPa, kPa, ff

₀₀

= = 22 MHz, MHz, ��p p = = 100 100 kPa kPa

bubble radius [-]r

stic e acous [Pa]pressurede [dB]amplitud

Time history of bubble radius and acoustic Time history of bubble radius and acoustic

pressure, and spectrum of the acoustic pressure pressure, and spectrum of the acoustic pressure

㻝㻡

(22)

Calculation Conditions Calculation Conditions

2.0 [ � m]

Initial Bubble Radius, R

_b0

293 [K]

Initial Temperature

101.3 [kPa]

Initial Ambient Pressure, p

_�0

1.34 [MHz]

Ultrasound Frequency, f

₀

sinus Waveform of Ambient Pressure

[ ]

q y,

₀

0 [kPa] ~ 1 [MPa]

Amplitude of Ambient Pressure

t f

Formula of ambient ultrasound pressure Formula of ambient ultrasound pressure

�

₀

�

⁵ ₀

0

sin 2 10

500

^�

�

� f t A f t � � p

p �

Bubble Radius Bubble Radius

Time history of bubble radius Time history of bubble radius

Bifurcation diagram of bubble radius Bifurcation diagram of bubble radius

㻝㻢

(23)

Acoustic Turbulence Acoustic Turbulence

Power spectrum of bubble radius

Power spectrum of bubble radius Power spectrum of acoustic pressure Power spectrum of acoustic pressure Power spectrum of bubble radius

Power spectrum of bubble radius Power spectrum of acoustic pressure Power spectrum of acoustic pressure

Experiment of PMs in Water

Experiment of PMs in Water (Depth = 30 m) (Depth = 30 m)

㻝㻣

(24)

Simulation Model Simulation Model

• Assumption

(1) Gases inside the bubble and the surrounding liquid move maintaining spherical symmetry.

(2) Pressure and temperature inside the bubble are (2) Pressure and temperature inside the bubble are

uniform except the thin boundary layer near the bubble wall.

(3) Non-condensable gas obeys Henry’s law at the bubble wall.

(4) Viscosity of the liquid is ignored except at the bubble wall.

Time History of Bubble Radius Time History of Bubble Radius

㻝㻤

(25)

Time History of Pressure Distribution

Time History of Pressure Distribution (p�=400 (p�=400 䌫䌫 Pa) Pa)

Time History of Pressure Distribution

Time History of Pressure Distribution (p�=400 (p�=400 䌫䌫 Pa) Pa)

㻝㻥

(26)

Time History of Emitted Pressure at 0.7 m Time History of Emitted Pressure at 0.7 m

Bubble Collapse and Shock Wave Formation Bubble Collapse and Shock Wave Formation

㻞㻜

(27)

⋡ᰴ

• 䈲䈛䉄䈮

න ᳇ᵃ䈱᜼േ

• න৻᳇ᵃ䈱᜼േ

• ᳇ᵃ䉪䊤䉡䊄䈱᜼േ

• ᵹ૕ᯏ᪾䈱䉨䊞䊎䊁䊷䉲䊢䊮䉣䊨䊷䉳䊢䊮 ᵹ૕ᯏ᪾ 䉨䊞䊁

• 䈍䉒䉍䈮䈍䉒䉍䈮

Model of a bubble cloud Model of a bubble cloud

The following phenomena are considered The following phenomena are considered The following phenomena are considered.

The following phenomena are considered.

�

� Compressibility of the liquid Compressibility of the liquid

�

� Evaporation and condensation Evaporation and condensation of the liquid at the bubble wall of the liquid at the bubble wall of the liquid at the bubble wall of the liquid at the bubble wall

�

� Evaporation and Evaporation and condensation of the mist condensation of the mist

�

� ff

condensation of the mist condensation of the mist inside the bubble

inside the bubble

�

� Heat transfer through Heat transfer through the bubble wall

the bubble wall

Shimada, Kobayashi and Matsumoto (1999)

㻞㻝

(28)

Assumptions Assumptions

Assumptions for a bubble cloud Assumptions for a bubble cloud

� The bubble cloud oscillates maintaining spherical symmetry.

� Bubbly liquid inside the cloud is treated as a continuum fluid.

� Bubbles move with the surrounding liquid.

� Coalescence and fragmentation of bubbles in the cloud are ignored.

� Viscosity of bubbly mixture is ignored in the cloud.

� Th t t f th li id i th l d i t t

� The temperature of the liquid in the cloud is constant.

� Each bubble oscillates maintaining spherical symmetry.

Assumptions for each bubble Assumptions for each bubble

� Each bubble oscillates maintaining spherical symmetry.

� The pressure and temperature inside the bubble are uniform except for the thin boundary layer near the bubble wall.

� Temperature at the bubble wall is equal to that of liquid

� Temperature at the bubble wall is equal to that of liquid.

� Mass of non-condensable gas inside a bubble is constant.

� Gases inside a bubble obey the van der Waals gas law.

� Coalescence and fragmentation of mist inside a bubble are ignored.

Governing equations 1 Governing equations 1

The motion of the bubble cloud interface

3 2 1

1 1 1 4

2 3

c c c c c

c c c w l

l c

R R R R d R

R R R p p

c c c c dt � R

� ^�

� �

� � � � � � � � � � � ��

� � � � � �

� � � � � ��

� � � �

��

The mass and momentum conservation equations

� �

2

�

²

� � �

1 1

1 0

l

r l lu

� � � �

� � � � � �

� 2�

� � �

l l

�

t r r �

� �

2

�

²

� �

²

�

1 1

1 0

l l

u p

r u

t r r r

� � � �

� � � � � ��

� � �

The conservation equation of the number density of bubbles

�

²

�

1 0

n r nu

� � � �

The governing equations for each bubble (The motion of the bubble wall,

� �

2 r nu_b 0

t�r r �

� �

g g q ( ,

The energy conservation equation, The nucleation rate equation of the mist

㻞㻞

(29)

Governing equations 2 Governing equations 2

The motion of the bubble wall (Fujikawa & Akamatsu equation)

3 2 4 4

1 2 1

2 3 3

l l

R m m R

RR R

c �c �c c

� � � �

� � � � �

� � � �

��

( j q )

, ,

1 2 0

2

l r R l r R

l l l l l l

p p Rp

mR R m m m

c c R c

� � � � � �

��

� � � �

� � � � ��

� � � �

� �

��

� �

2

2 4

vi gi l l

l R

m m

p p p � ^�� ^�� R �

� � � � � � ��

� �

, 2 4

l r R v g

b l

l vi gi

p p p R

R R �

� � �

� � � ��

The energy conservation equation in gas phase with mist

� �

2 2

g g g v v v

vg g vv v vl c

g v

p M d p M d C M C M C M dT

dt dt dt

dM T dM dM dM

� �

� � � �

� � �

0

c v c c

v l

r R

dM T dM dM dM

L S h h

dt � r dt dt dt

�

� � �

� � � � � ��

The nucleation rate equation of mist q

Frequency response of a bubble cloud Frequency response of a bubble cloud

Maximum pressure inside the bubbles in the cloud Maximum pressure inside the bubbles in the cloud

㻞㻟

(30)

Natural modes (

Natural modes (�� pp = 10 kPa) = 10 kPa)

ff = 200 kHz = 200 kHz ff = 600 kHz = 600 kHz Water pressure inside the bubble cloud

Water pressure inside the bubble cloud Natural mode shapes

Natural mode shapes

Linearized analysis of a spherical bubble Linearized analysis of a spherical bubble cloud (d’Agostino & Brennen, 1989) cloud (d’Agostino & Brennen, 1989)

Natural mode shapes Natural mode shapes

( g , )

•• Continuity equation Continuity equation

•• Momentum equation Momentum equation R l i h

R l i h Pl Pl t t ti ti

•• Rayleigh Rayleigh--Plesset equation Plesset equation

Frequency response of a bubble cloud Frequency response of a bubble cloud

Maximum pressure inside the bubbles in the cloud Maximum pressure inside the bubbles in the cloud

㻞㻠

(31)

Pressure wave in the bubble cloud Pressure wave in the bubble cloud

��pp = 75 kPa, 190 kHz = 75 kPa, 190 kHz

��pp = 10 kPa, 200 kHz = 10 kPa, 200 kHz

Collapse of the cloud

Collapse of the cloud ((�� pp = 125 kPa, 160 kHz) = 125 kPa, 160 kHz)

a]sure [MPapress

/R [ ]

time, t/ T [-]

r/ R_c0 [-]

Water pressure inside the bubble cloud Water pressure inside the bubble cloud

㻞㻡

(32)

⋡ᰴ

• 䈲䈛䉄䈮

න ᳇ᵃ䈱᜼േ

• න৻᳇ᵃ䈱᜼േ

• ᳇ᵃ䉪䊤䉡䊄䈱᜼േ

• ᵹ૕ᯏ᪾䈱䉨䊞䊎䊁䊷䉲䊢䊮䉣䊨䊷䉳䊢䊮 ᵹ૕ᯏ᪾ 䉨䊞䊁

• 䈍䉒䉍䈮䈍䉒䉍䈮

䉪䊤䉡䊄䉨䊞䊎䊁䊷䉲䊢䊮䈱᷵᡼಴

㻞㻢

(33)

⠢೉䉨䊞䊎䊁䊷䉲䊢䊮䈱ᤨ㑆⊒ዷ

Cavity Velocity(|u|) Pressure

6 . 0 deg,

6 �

�

� �

�

⢛ ᥊

䊘䊮䊒䈱↢↥䉮䉴䊃ૐᷫ � ዊဳൻ

㜞ㅦ䋨㜞࿁ォ䋩ൻ

䉨䊞䊎䊁䊷䉲䊢䊮䉣䊨䊷䉳䊢䊮䉨䊞䊎䊁䉲䊢䊮䉣䊨䉳䊢䊮

䉣䊨䊷䉳䊢䊮੍᷹䋺 ᓥ᧪䈲ታ㛎⊛䈭ᚻᴺ䈏ਥᵹ

CFD

䈱᦭ലᵴ↪

�

⹜㛎䈱⋭ജൻ䈮䉋䉎㐿⊒ᦼ㑆⍴❗

䉨䊞䊎䊁䊷䉲䊢

䊮䉨䊞䊎

䉨䊞䊎䊁䊷䉲䊢䊮㐳䈘䊁

䉣䊨䊷䉳䊢䊮੍᷹ᴺ䈱৻଀䋺

นⷞൻ⸘᷹

䊁䊷䉲䊢䊮㐳䈘

䉣䊨䉳䊮ㅦᐲ䉕੍᷹

㪚㪝㪛੍᷹

ታ㛎ᑼ

䉣䊨䊷䉳䊢䊮

䉣䊨䊷䉳䊢䊮ㅦᐲ䉕੍᷹

㻞㻣

(34)

⋡ ⊛

᳇ᵃ䊝䊂䊦⸃ᨆ䉮䊷䊄

䉨䊞䊎䊁䊷䉲䊢䊮᳇ᵃ䈱፣უ

䊶᳇ᵃᢙኒᐲಽᏓ

᳇ᵃ䈱⹦⚦᜼േ

᳇ᵃ䊝䊂䊦⸃ᨆ 䊄

᳇ᵃ䈱፣უ 䋨ᶖṌ䋩

䊶᳇ᵃ䈱⹦⚦᜼േ

䋨᳇ᵃ䈱ਗㅴ䊶૕Ⓧㆇേ䋩

᳇ᵃ 䉴䊌䉟䉪⁁䈱 ⴣ᠄࿶⊒↢

㆙ᔃ䊘䊮䊒

ⴣ᠄࿶⊒↢

䉨䊞䊎䊁䊷䉲䊢䊮ᒝ䈘䉨䊞䊁䉲䊢䊮ᒝ䈘䉣䊨䊷䉳䊢䊮⊒↢૏⟎䊶㊂

䋨ታ㛎䌄䌂䋩

䉳䊢䊮⊒↢૏⟎ ㊂

⸘▚ᚻᴺ

(1) ઒ቯ ( )

ᶧ⋧䋺

᳓ (

㕖࿶❗ᕈ

)

᳇⋧䋺

᳇ᵃ (

࿶❗ᕈ

)

䊶᳇ᵃ䈲⃿ᒻ䈪ว૕䉇ಽⵚ䈭䈚

᳇ᵃౝ䈲⫳᳇䈍䉋䈶ਇಝ❗䉧䉴䊶᳇ᵃౝ䈲⫳᳇䈍䉋䈶ਇಝ❗䉧䉴

䋨╬᷷⤘ᒛ䊶ᢿᾲ෼❗䋩

䊶᳇⋧䈱ኒᐲ䉇ㆇേ㊂䈲ᶧ⋧䈮 Ყ䈼䈩ᓸዊ

( ) ᡰ㈩ᣇ⒟ᑼ (2) ᡰ㈩ᣇ⒟ᑼ

䊶ᶧ⋧૕Ⓧ₸䈱଻ሽᑼ 䊶᳇ᵃᵹ䈱ㆇേ㊂଻ሽᑼ 䊶᳇ᵃᵹ䈱ㆇേ㊂଻ሽᑼ 䊶᳇ᵃᢙኒᐲ䈱଻ሽᑼ

䊶࿶ജᣇ⒟ᑼ䋨૕Ⓧ₸䈱᜔᧤᧦ઙ䈎䉌ዉ಴䇯ᡆૃ࿶❗ᕈᴺ䈱⠨䈋ᣇ䈮ၮ䈨䈒䋩

㻞㻤

(35)

ᡰ㈩ᣇ⒟ᑼ -1

G H F E G F E t

Q ˆ ˆ ˆ ˆ ˆ

_v

ˆ

_v

ˆ

_v

ˆ

� �

�

� �

�

� �

�

� �

�

� �

�

� �

�

� _� _� _� _� _� --- (1)

�

u f f

L L L

L

� �

�

� �

�

� f u U p

U f

x L L L L

L L

� �

�

� �

�

� �

�

_x _xx _y _xy _z _xz

0 � �

�

� �

�

� � � � �

�

� w J

f v f

u f Q

L L L

ˆ /

� �

�

� �

�

� �

�

p J U w f

p U v f

p f

E

z L L L L

y L L L L

x L L L L

ˆ /

� �

�

� �

�

� �

�

J E

zz z zy y zx x

yz z yy y yx x

xz z xy y xx x

v

/

ˆ

� �

�

� �

�

� �

�

n p

G

� � �

� �

� n U

U f c U f c

G G

G G L L L L

2 2

� �

�

� �

�

� �

�

0 0

� �

�

� �

�

u v

L L

0 � �

� �

�

࿁ォ⋥੤ᐳᮡ♽䈮䈍䈔䉎⛘ኻㅦᐲᚑಽ

J r n D r c

u H

G G G G L

L L

/ 4

ˆ 0

2 2

� � � � �

� �

�

ᄖജ㗄䈮䉮䊥䉥䊥ജ

t

n D r c

G G G L

0 4

� �

�

� �

�

ᡰ㈩ᣇ⒟ᑼ -2

R l i h Pl t ᑼ

᳇ᵃ䈱૕Ⓧㆇേ

̪Ꮢ⽼䉮䊷䊄╬䈪䈲Ꮐㄝ╙䋱㗄䉕⋭⇛䈚䇮◲ᤃᑼ䉕⸃䈒

䋨᳇ᵃ᜼േ ᭎⇛䉕⸃䈒䋩䉅䈏䈅䉎

Rayleigh-Plesset ᑼ

) )(

4 ( ) 1

2 (

3

2

2 2

Gi Li Gi Li L

B G

G

p p u u u u

D Dr r

r D � � � � � � --- (2)

䋨᳇ᵃ᜼േ䈱᭎⇛䉕⸃䈒䋩䉅䈱䈏䈅䉎䇯

) )(

4 ( )

2 (

2 Li Gi Li Gi

L

G

Dt Dt �

D Dr p T

p

_B _G _v

1

^G

2 � 4 �

�

᳇ᵃᓘ

( ) --- (3)

Dt r p r

p p

G G

v G

B

� _{( )}

䊗䉟䊄₸ ᳇ᵃ䈱ਗㅴㆇേ

᳇ᵃౝ࿶ജ

䊗䉟䊄₸

G G

G

r n

f

³

3 4 �

�

᳇ᵃ䈱ਗㅴㆇേ

� 0

�

_pi _Di _Li _Ci

Ai

F F F F

--- (4) F --- (5)

3 F

_Ai

: ઃടᘠᕈജ

F

_pi

: ๟࿐ᵹ૕䈱ടㅦ䈮䉋䉎ജ

᛫ജ䈍䈶឴ജ

᳇ᵃᢙኒᐲ

F

_Di

, F

_Li

: ᛫ജ䈍䉋䈶឴ജ F

_Ci

: 䉮䊥䉥䊥ജ

㻞㻥

(36)

㪀㩷⸃ᨆ㗔ၞ䈫Ⴚ⇇᧦ઙ

౉ญႺ⇇

㫑

ๆㄟ஥

౉ญႺ⇇

䊶ᵹㅦ䇮᳇ᵃᓘ䇮᳇ᵃᢙኒᐲ╬䈱৻᭽ᵹ౉

⋥౞▤ᵹ ಴ญႺ⇇

〝 ฯ಴஥

ᚸᒻ᜛ᄢᵹ〝

Ⴚ⇇

䊶࿶ജ৻ቯ 䉲䊠䊤䉡䊄

ᚸᒻ᜛ᄢᵹ〝

䊶䉲䊠䊤䉡䊄䈅䉍䊶⠀ᩮᨎᢙ䋶ᨎ

㪛㪔㩷㪊㪇㪉㩷㫄㫄㪈㪏㪇㩷㫄㫄㪧㪪

䊶⸃ᨆᩰሶᢙ ⚂ 㪉㪉㪇㪃㪇㪇㪇

㪪㪪

㪉㪌㪇㩷㫄㫄䊶ೋᦼ᧦ઙ

䊗䉟䊄₸ 㪑㩷㪇㪅㪇㪇㪈㩷㩿㪔㩷㪇㪅㪈䋦㪀

᳇ᵃᓘ 㪑㩷㪈㪅㪇㬍㪈㪇^㪄㪌㫄㩿㪈㪇㪀

㩿㪔㩷㪈㪇㩷㱘㫄㪀

ᱜ࿶㕙䋨㪧㪪䋩䈫⽶࿶㕙䋨㪪㪪䋩

䈱㑆䈱㪈⠢㑆

⸃ᨆ⚿ᨐ䈍䉋䈶⠨ኤ 䋱

4 1 䉨䊞䊎䊁䊷䉲䊢䊮ᕈ⢻

⸃ᨆ⚿ᨐ䈲䇮䊘䊮䊒䈱৻⥸⊛䈭䉨䊞䊎䊁䉲䊢䊮ᕈ⢻䈫

4.1 䉨䊞䊎䊁䊷䉲䊢䊮ᕈ⢻

ㇱಽ⽶⩄᧦ઙ

( Q/Q

d

= 0.6)

䉨䊞䊎䊁䊷䉲䊢䊮ᕈ⢻䈫 ቯᕈ⊛䈮৻⥌

)

1.4 1.6

1.2 1.4

0 0.2 0.4 0.6 0.8 1

NPSH’^{R cal}= 0.068

0 8 1.0 1.2

�

^0.8

Case 1 1

Ȁ

0.4 0.6

�

0.8

0.4 0.6

Exp.

NPSH’^{R exp}= 0.094

Ȁ

� ₌ H / ( U

_t²

/ 2g )

NPSH’ = NPSH / ( U

_t²

/ 2g)

0.0 0.2

0.0 0.2 0.4 0.6 0.8 1.0

0

p 0.2

Cal.

0.0 0.2 0.4 0.6 0.8 1.0

NPSH'

㻟㻜

(37)

⸃ᨆ⚿ᨐ䈍䉋䈶⠨ኤ 䋲䊗䉟䊄₸䈫᳇ᵃᢙኒᐲ

䉨䊞䊎䊁䊷䉲䊢䊮㗔ၞ ᳇ᵃᩭ䈱㓸Ⓧ

䉨䊞䊎䊁䉲䊢䊮㗔ၞ ᳇ᵃᩭ䈱㓸Ⓧ

㩿㫄㩷㩷㪀^㪄㪊

ォᣇะ ࿁ォᣇะ

࿁ォᣇะ ࿁ォᣇะ

䊗䉟䊄₸ ᳇ᵃᢙኒᐲ

⸃ᨆ⚿ᨐ䈍䉋䈶⠨ኤ 䋳

㪇㪅㪈㪋㪇㪅㪈㪍

㪇㪇

㪇㪇㪇㪇㪇㪈㪈㪈㪈㪈㪈

䊗䉟䊄₸

L.E. T.E.

㪇㪅㪇㪏㪇㪅㪈㪇㪇㪅㪈㪉㪇㪅㪈㪋

㫉㪸㪺㫋㫀㫆㫅㪃㩷㪽㪞

㪇㪇

䊗䉟䊄₸ 㪇

s s =0 L.E.

㪇㪅㪇㪉㪇㪅㪇㪋㪇㪅㪇㪍

㪭㫆㫀㪻㩷㪝㫉

㪇㪇㪇

s =1

^࿁ォᣇะ _㪇㪅㪇㪇

㪇㪅㪇㪇㪅㪈㪇㪅㪉㪇㪅㪊㪇㪅㪋㪇㪅㪌㪇㪅㪍㪇㪅㪎㪇㪅㪏㪇㪅㪐㪈㪅㪇㪛㫀㫊㫋㪸㫅㪺㪼㪃㩷㫊

㪇

㪈㪉㪇

㫅㪞㪈㪋㪅㪇

㪈㪉㪈㪋

㪇㪇㪇㪇㪇㪈㪈㪈㪈㪈㪈

᳇ᵃᢙኒᐲ

T.E.

㪍㪇㪏㪅㪇㪈㪇㪅㪇㪈㪉㪅㪇

㪹㪼㫉㩷㪛㪼㫀㫅㫊㫀㫋㫐㪃㩷㪍

㪏㪈㪇

᳇ᵃᢙኒᐲ

⠀ᩮ⽶࿶㕙䈮ᴪ䈦䈢

⠀ᩮ೨✼䈎䉌䈱〒㔌㫊

㪇㪇㪉㪅㪇㪋㪅㪇㪍㪅㪇

㪙㫌㪹㪹㫃㪼㩷㪥㫌㫄㪹

㪇㪉㪋㪍

G G

G

r n

f � 4 �

³ _㪇㪅㪇

㪇㪅㪇㪇㪅㪈㪇㪅㪉㪇㪅㪊㪇㪅㪋㪇㪅㪌㪇㪅㪍㪇㪅㪎㪇㪅㪏㪇㪅㪐㪈㪅㪇㪛㫀㫊㫋㪸㫅㪺㪼㪃㩷㫊

G 㪇 G

f

G

3

䊗䉟䊄₸ ᳇ᵃᢙኒᐲ

㻟㻝

宇宙航空研究開発機構特別資料JAXA Special Publication