JAXA-SP indd

ᅛẼ┦㐃ᡂၥ㢟࡟࠾ࡅࡿ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫࣔࢹࣝࡢᵓ⠏࡜㧗㏿ὶゎᯒ࡬ࡢᛂ⏝

㯮 ༟ྖ㸪ఫ 㝯༤ -$;$㸪㫽ྲྀ኱Ꮫ

Construction of Interface model in Solid-fluid Interaction Problem and Application

to High Speed Flow Analysis

by

Takuji Kurotaki (JAXA) and Takahiro Sumi㸦Tottori University㸧

ABSTRACT

A new interface model in solid-fluid interaction problem is presented. A level set function is used for the definition of shapes and flow properties are corrected within three layers of stencils around interface. This approach is very simple and robust and can capture the detail of flow structures including discontinuities such as shock waves and slip lines etc.. Some basic important 2-D Euler flow problem are solved to verify effects of this approach with WCNS (Weighted Compact Nonlinear Scheme) including a new type of compact scheme to improve robustness including the transonic flow around a 2-D airfoil, the supersonic duct flow around a prism and a circular cylinder and the moving-shock/obstacle interaction problem (Schardin’s problem). The extension of this method to the moving problem with body deformation is straightforward. 㸯㸬ࡣࡌࡵ࡟ ὶయ୰࡟⢏Ꮚ⩌ࢆྵࡴ࠸ࢃࡺࡿᅛẼ┦㐃ᡂၥ㢟ゎᯒࡣࠊ ΰ┦ὶၥ㢟ࡢ㔜せ࡞࢝ࢸࢦ࣮࡛ࣜ࠶ࡾࠊ≉࡟ప㏿ὶ࡟࠾ࡅ ࡿ᭱㏆ࡢⓎᒎࡣⴭࡋ࠸(1)_{ࠋࡇࡢ㡿ᇦ࡛ࡣࠊ┠ⓗ࡟ᛂࡌ࡚ࠊ} ࠸ࡃࡘ࠿ࡢゎᯒ࢔ࣉ࣮ࣟࢳࡀᏑᅾࡍࡿࡀࠊࡑࡢ୰࡛ࡶࠊ㸯 ಶ㸯ಶࡢ⢏Ꮚ࿘ࡾࡢὶࢀࢆ⢭ᐦ࡟ゎࡁࠊ඲యࡢὶࢀࢆ᫂ࡽ ࠿࡟ࡍࡿ DEM-DNS ἲࢆࡣࡌࡵ࡜ࡍࡿ┤᥋ィ⟬ἲࢆ⏝࠸ࡓ ᪉ἲㄽࡀ᭱㏆ὀ┠ࢆ㞟ࡵ࡚࠸ࡿ(1)_{ࠋࡋ࠿ࡋࠊ㧗㏿ὶࡢศ㔝} ࡛ࡣࠊ௵ពᙧ≧ࡢ⢏Ꮚࢆ࢜࢖ࣛࣜ࢔ࣥⓗ࡟ᤕࡽ࠼ࠊ࠿ࡘ⾪ ᧁἼ➼ࡢ୙㐃⥆⌧㇟࡜ࡢ┦஫స⏝ࢆ⢭ᐦ࡟ゎࡃᚲせࡀ࠶ࡿ ࡓࡵ࠿ࠊᮍࡔⓎᒎ㏵ୖࡢឤࡀ࠶ࡿࠋ ୍᪉ࠊᚑ᮶ࡼࡾ㏻ᖖ⾜ࢃࢀ࡚࠸ࡿࠊ౛࠼ࡤື࠿࡞࠸⩼➼ ࡢ௵ព≀య࿘ࡾࡢὶࢀゎᯒࡣࠊᅛయ࡜Ẽయࡢ㐃ᡂၥ㢟ࡢ᭱ ࡶࢩࣥࣉࣝ࡞ࢣ࣮ࢫ࡜ࡳ࡞ࡍࡇ࡜ࡀ࡛ࡁࡿࠋ⌧ᅾࡲ࡛࡟ࠊ ከࡃࡢ IBM(Immersed boundary method)࡟㛵ࡍࡿᡭἲࡀᥦ᱌ ࡉࢀࠊ▴ᙧ᱁Ꮚࢆ⏝࠸࡚௵ពᙧ≧࿘ࡾࡢゎᯒࡀྍ⬟࡟࡞ࡾ ࡘࡘ࠶ࡿࠋࡋ࠿ࡋࠊᙉ࠸⾪ᧁἼࢆྵࡴ㧗㏿ὶ࡟ᑐࡍࡿ㐺⏝ ࡣᮍࡔࢳࣕࣞࣥࢪࣥࢢ࡞ㄢ㢟࡛࠶ࡾࠊ◊✲౛ࡶከࡃࡣ↓ࡃ (2)_{ࠊ౫↛Ⓨᒎࡢవᆅࢆṧࡋ࡚࠸ࡿࠋ} ᮏ◊✲࡛ࡣࠊ㧗㏿ὶࢆᑐ㇟࡜ࡋࡓᅛẼ┦㐃ᡂၥ㢟࡟ᑗ᮶ ⓗ࡟㐺⏝ྍ⬟࡛ࠊ࠿ࡘ⡆౽࡞࢖ࣥࢱ࣮ࣇ࢙࣮ࢫࣔࢹࣝࢆᥦ ᱌ࡍࡿࠋ௒ᅇࡣࠊ➨୍ሗ࡜ࡋ࡚ࠊࢩࣥࣉࣝ࡞≀య࿘ࡾࡢ㧗 ㏿ὶࢀゎᯒ࡟㐺⏝ࡋࠊᛶ⬟ࡢホ౯ࢆヨࡳࡿࠋᮏ᪉ἲࡣࠊᚋ ㏙ࡍࡿࡼ࠺࡟ࠊ≉ᐃࡢゎᯒἲࢆ㑅ࡤ࡞࠸ỗ⏝ⓗ࡞ᡭἲ࡛࠶ ࡿࡀࠊᮏ◊✲࡛ࡣࠊᡃࠎࡀᥦ᱌୰ࡢࣟࣂࢫࢺᛶࢆྥୖࡉࡏ ࡓ㔜ࡳ௜ࢥࣥࣃࢡࢺࢫ࣮࣒࢟㸦WCNS ἲ㸧ࢆ࣮࣋ࢫ࡟ࡋࡓ 㸳ḟ⢭ᗘࡢ㧗⢭ᗘゎἲ(3)ࠊ(4)_{࡜⤌ࡳྜࢃࡏࡓ⤖ᯝࢆሗ࿌ࡍࡿࠋ} 㸰㸬ᇶ♏᪉⛬ᘧ Ẽయࡣ⌮᝿Ẽయࢆ௬ᐃࡋࠊ௨ୗࡢ࢜࢖࣮ࣛ᪉⛬ᘧ⣔ࢆᇶ ♏᪉⛬ᘧ࡜ࡍࡿࠋ μ“ μ–൅ ෍ μ 욌  Œൌͳ ݂௝ሺ“ሻൌͲ    ሺͳሻ ݍ ൌ ሺߩǡ ߩݑଵǡήήήǡ ߩݑேǡ ܧሻ்       ሺʹሻ ݂௝ൌ ൫ߩݑ௝ǡ ߩݑଵݑ௝൅ ݌ߜ௝ǡଵǡήήήǡ ߩݑேݑ௝ ൅ ݌ߜ௝ǡேǡ ሺܧ ൅ ݌ሻݑ௝൯ ் ሺ͵ሻ ܧ ൌ ߩ݁ ൅ ߩ࢛ ή ࢛Ȁʹ  ሺͶሻ ࡇࡇ࡛ࠊN ࡣḟඖᩘࠊu ࡣ㏿ᗘ࣋ࢡࢺࣝࠊujࡣ j ḟඖ┠ࡢ㏿ ᗘᡂศࠊp ࡣᅽຊࠊE ࡣ඲࢚ࢿࣝࢠ࣮ࠊe ࡣ༢఩㉁㔞ᙜࡓࡾ ࡢෆ㒊࢚ࢿࣝࢠ࣮࡛࠶ࡿࠋ ィ⟬᱁Ꮚࡣ▴ᙧᙧ≧࡜ࡋࠊ≀యᙧ≧ࡣᚋ㏙ࡍࡿࡼ࠺࡟ࠊ ࣞ࣋ࣝࢭࢵࢺ㛵ᩘIࢆ౑⏝ࡋ࡚ᐃ⩏ࡍࡿࠋࡓࡔࡋࠊᮏሗ࿌ ࡛ࡣࠊ≀యᙧ≧ࡣኚᙧࡋ࡞࠸࡜௬ᐃࡍࡿࡓࡵࠊI࡟㛵ࡍࡿ ⛣ὶ᪉⛬ᘧࡣ⪃៖ࡋ࡞࠸ࠋ 㸱㸬ゎᯒᡭἲ 㸱㸬㸯 ᩘ್ゎᯒࢫ࣮࣒࢟ࡢᴫせ ᮏ◊✲࡛ࡣࠊᇶ♏᪉⛬ᘧ (1) ᘧࢆ㸳ḟ⢭ᗘ㔜ࡳ௜ࢥࣥࣃ ࢡࢺࢫ࣮࣒࢟(Weighted Compact Nonlinear Scheme: WCNS ἲ) ࡛ゎࡃࡇ࡜ࢆ⪃࠼ࡿ(5), (6)_{ࠋࡇࡇ࡛ࡣࠊ⡆༢ࡢࡓࡵࠊḟ} ࡢ㸯ḟඖ཮᭤ᆺಖᏑ᪉⛬ᘧࢆ⪃࠼ࡿࠋ μ“ μ–൅ μ μš݂ሺ“ሻൌͲ    ሺͷሻ ࡇࡇ࡛ࡣࠊࡍ࡭࡚ࡢ T ࡟ᑐࡋ࡚݂݀ሺݍሻȀ݀ݍ ൒ Ͳ࡜௬ᐃࡍࡿࠋ ᘧࡣࠊ‽㞳ᩓ໬ࡍࡿࡇ࡜࡟ࡼࡾࠊ௨ୗࡢᙧ࡛ホ౯ࡉࢀ ࡿࠋ ߲ݍ ߲ݐฬ௜ ൌǦ݂ᇱ_{    ሺ͸ሻ} ࡇࡇ࡛ࠊq ࡣࢫ࣮࢝ࣛኚᩘࠊi ࡣᗙᶆ࢖ࣥࢹࢵࢡࢫࢆ⾲ࡍࠋ ⥆࠸࡚ࠊྛࢭࣝ㛫ࡢࣇࣛࢵࢡࢫࢆ㸱✀㢮ࡢࢧࣈࢫࢸࣥࢩࣝ ࡟࠾ࡅࡿ㸱ḟ⢭ᗘࣛࢢࣛࣥࢪ࢙⿵㛫ࡢฝ⤖ྜ࡟ࡼࡾࠊ㠀⥺ ᙧࡢ㸳ḟ⢭ᗘ⿵㛫ࢆ⪃࠼ࡿࠋ ݂ҧ௜ାଵȀଶൌ ෍ ߱௞ ʹ ൌͲ ݂ҧ௞ǡ௜ାଵȀଶ   ሺ͹ሻ ࡇࡇ࡛ࠊࡑࢀࡒࢀࡢపḟ⢭ᗘ⿵㛫ࡣࠊ ݂ҧ௞ǡ௜ାଵȀଶൌ݂௜൅ οݔ ʹ ׎௞ǡ௜ ሺଵሻ_ ൅ሺοݔሻ ଶ ͺ ׎௞ǡ௜ ሺଶሻ  ሺͺሻ ࡛୚࠼ࡽࢀࠊᘧ୰ࡢ㸯㝵ᚤศ࡜㸰㝵ᚤศࡣࠊ ۏ ێ ێ ێ ۍ׎଴ǡ௜ ሺଵሻ ׎_ଵǡ௜ሺଵሻ ׎_ଶǡ௜ሺଵሻےۑ ۑ ۑ ې ൌ ͳ ʹοݔ቎ ݂௜ିଶǦͶ݂௜ିଵ൅ ͵݂௜ Ǧ݂௜ିଵ൅ ݂௜ାଵ Ǧ͵݂௜൅Ͷ݂௜ାଵǦ݂௜ାଶ ቏൅ሺሺοݔሻଶ_{ሻሺͻሻ} ۏ ێ ێ ێ ۍ׎଴ǡ௜ ሺଶሻ ׎_ଵǡ௜ሺଶሻ ׎_ଶǡ௜ሺଶሻ_ےۑ ۑ ۑ ې ൌ ͳ ሺοݔሻଶ቎ ݂௜ିଶǦʹ݂௜ିଵ൅ ݂௜ ݂௜ିଵǦʹ݂௜൅ ݂௜ାଵ ݂௜Ǧʹ݂௜ାଵ൅ ݂௜ାଶ ቏൅ሺοݔሻ ሺͳͲሻ ࡛࠶ࡿࠋ㠀⥺ᙧ㔜ࡳࡣࠊ௨ୗ࡛୚࠼ࡽࢀࡿ(5)ࠋ

Construction of Interface model in Solid-fluid Interaction Problem and Application

to High Speed Flow Analysis

by

Takuji Kurotaki (JAXA) and Takahiro Sumi㸦Tottori University㸧

ABSTRACT

A new interface model in solid-fluid interaction problem is presented. A level set function is used for the definition of shapes and flow properties are corrected within three layers of stencils around interface. This approach is very simple and robust and can capture the detail of flow structures including discontinuities such as shock waves and slip lines etc.. Some basic important 2-D Euler flow problem are solved to verify effects of this approach with WCNS (Weighted Compact Nonlinear Scheme) including a new type of compact scheme to improve robustness including the transonic flow around a 2-D airfoil, the supersonic duct flow around a prism and a circular cylinder and the moving-shock/obstacle interaction problem (Schardin’s problem). The extension of this method to the moving problem with body deformation is straightforward.

㸦Tottori University㸧 JAXA，　　鳥取大学

߱_௞௃ௌൌ ߙ௞ ௃ௌ σଶ ߙ_௟௃ௌ ௟ୀ଴ ǡߙ_௞௃ௌൌ ߱തതതത௞ ሺߚ_௞൅ ߳ሻ௣ ሺͳͳሻ ࡇࡇ࡛ࠊ'x ࡣ᱁Ꮚ㛫㝸ࠊಀᩘ ߱തതതതǡ ߱଴തതതതǡ ߱ଵതതതത ࡣ᭱㐺㔜ࡳࠊp ࡣᩚᩘ㸦ࡇଶ ࡇ࡛ࡣࠊ ࡜ࡋࡓ㸧ࠊHࡣᚤᑠಀᩘࢆ⾲ࡍࠋࡲࡓࠊ smoothness indicator Ekࡣḟᘧ࡛୚࠼ࡽࢀࡿࠋ ߚ௞ൌ ෍ሺοݔሻଶ௟ିଵන ቆ ݀௟_݂ҧ ௞ ݀ݔ௟ቇ ଶ ௫ ೔శభ_మ ௫ ೔షభ_మ ʹ Žൌͳ ݀ݔ ൌ ቀሺοݔሻ׎_௞ǡ௜ሺଵሻቁଶ൅ଵଷ ଵଶቀሺοݔሻ ଶ_׎ ௞ǡ௜ ሺଶሻ ቁଶሺͳʹሻ ࡇࡢࡼ࠺࡟ᵓᡂࡉࢀࡓᘧࢆ⏝࠸࡚ࠊᘧࡢ݂ᇱ_ࡣࠊⴭ ⪅ࡽ࡟ࡼࡾ᪂ࡋࡃᥦ᱌ࡉࢀࡓࠊ୰Ⅼࡢ⿵㛫್࡜ࣀ࣮ࢻⅬࢆ ⏝࠸ࡓ௨ୗࡢ㸴ḟ⢭ᗘ୕㔜ᑐゅ୰ᚰࢥࣥࣃࢡࢺᕪศࢫ࣮࢟ ࣒ࢆ⏝࠸࡚ồࡵࡽࢀࡿ(3)ࠋ Ƚ݂௜ିଵᇱ ൅݂௜ᇱ൅Ș݂௜ାଵᇱ   ͳ οݔ෍ ൤ܽଶ௞ିଵ൬݂ҧ௜ା௞ିଵ_ଶെ ݂ҧ௜ି௞ାଵ_ଶ൰ ൅  ܽଶ௞ሺ݂௜ା௞െ ݂௜ି௞ሻ൨ ୩  ൅ሺሺοݔሻ଺_{ሻሺͳ͵ሻ} ሺܽଵǡ ܽଶǡ ܽଷሻ ൌ ቀ ଽିଶ଴ఈ ଺ ǡ ିଽା଺ଶఈ ଷ଴ ǡ ଵାଵଶఈ ଷ଴ ቁሺͳͶሻ ᘧࡢࢥࣥࣃࢡࢺࢫ࣮࣒࢟࡟࠾࠸࡚ࠊDࡢᐇ⏝ⓗ࡞್ࡢ ⠊ᅖࡣ࡛࠶ࡾࠊᚑ᮶ࡢࢫ࣮࣒࢟࡟ẚ࡭ :&16 ἲ ࡢሀ∼ᛶࡀ᱁ẁ࡟ྥୖࡍࡿࠋ ᮏゎἲࡢ࢜࢖࣮ࣛ᪉⛬ᘧ࡬ࡢ㐺⏝࡟㝿ࡋ࡚ࡣ㸪ཎጞኚᩘ ࢆ≉ᛶኚᩘ࡬ኚ᥮ࡋ࡚㠀⥺ᙧ⿵㛫ࢆ⾜࠸㸪ࡑࡢᚋ⿵㛫್ࢆ ෌ᗘཎጞኚᩘ࡬㏫ኚ᥮ࡋ࡚㸪㏆ఝ࣮࣐ࣜࣥゎἲ࡟ࡼࡿᩘ್ ὶ᮰ࢆィ⟬ࡋ࡚࠸ࡿ㸬≉ᛶኚᩘ⿵㛫࡟ࡘ࠸࡚ࡣ౛࠼ࡤ㸪ᩥ ⊩ (7) ࢆཧ↷ࡉࢀࡓ࠸㸬ࡲࡓࠊ᫬㛫✚ศࢫ࣮࣒࢟࡟ࡣ 3 ḟ⢭ ᗘ TVD ࣝࣥࢤࢡࢵࢱἲ (5) _{ࢆ⏝࠸ࡓࠋ}  㸱㸬㸰 ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫ㠃㏆ഐࡢྲྀࡾᢅ࠸ ᮏゎᯒἲ࡛ࡣࠊࣞ࣋ࣝࢭࢵࢺ㛵ᩘࢆ⏝࠸࡚ᐃ⩏ࡉࢀࡓ࢖ ࣥࢱ࣮ࣇ࢙࣮ࢫ㏆ഐ࡟࠾࠸࡚ࠊ≀య࡟᭱ࡶ㏆࠸᱁ᏊⅬ⩌࠿ ࡽ࡞ࡿ㸯ᒙ࡜≀యෆ㒊ࡢ㸰ᒙࡢィ㸱ᒙศࡢ≀⌮㔞࡟ᑐࡋ࡚ ಟṇࢆ᪋ࡋࠊ≀యෆ㒊ࡶ௬᝿ⓗ࡞ὶయ࡜ࡋ࡚ᢅ࠺ࡇ࡜ࢆ⪃ ࠼ࡿࠋ ᭱ࡶᇶᮏⓗ࡞㸰ḟඖၥ㢟ࡢࢣ࣮ࢫ࡜ࡋ࡚ࠊFig. 1 ࡟♧ࡍ ࡼ࠺࡞ࠊ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫ㠃㸦I=0㸧ࡀ㸰ḟඖ᱁ᏊⅬ࢖ࣥ ࢹࢵࢡࢫ j ࡜ j-1 ࡢ㛫࡟Ꮡᅾࡍࡿሙྜࢆ⪃࠼ࡿ࡜ࠊ( i, j )ࠊ ( i, j-1 )ࠊ( i, j-2 )ࡢྛ᱁ᏊⅬ࡟࠾ࡅࡿ≀⌮㔞ࡣ௨ୗࡢࡼ࠺࡟ ಟṇࡉࢀࡿࠋ (a) ( i, j )Ⅼ㸸 ㏿ᗘ࣋ࢡࢺࣝ ݑሬԦൌ൫ݑ௜ǡ௝ǡ ݒ௜ǡ௝൯ ࢆݑሬሬሬԦൌ൫ݑᇱ ᇱ௜ǡ௝ǡ ݒᇱ௜ǡ௝൯࡟ಟṇࡍࡿࠋ ࡓࡔࡋࠊݑ_{ሬሬሬԦࡣ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫ㠃࡟࠾ࡅࡿ᥋⥺࣋ࢡࢺ࡛ࣝ}ᇱ ࠿ࡘȁݑሬԦȁൌหݑ_{ሬሬሬԦหࠋ} ᇱ (b) ( i, j-1 )Ⅼ㸸 ( i, j-1 )Ⅼ࠿ࡽ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫ㠃࡟ᆶ┤᪉ྥ࡟ᘬ࠸ࡓ┤⥺ ࡜( i-1, j )ࠊ( i, j )ࠊ( i+1, j )ࡢྛ᱁ᏊⅬ࡜ࡢ㊥㞳 s1ࠊs2ࠊs3ࢆ ᐃ⩏ࡋࠊࡇࡢᆶ⥺ࡀ j 㠃࡟࠾࠸࡚( i, j )Ⅼࡢ࡝ࡕࡽഃࢆᶓษ ࡿ࠿࡟ࡼࡗ࡚ࠊs1ࠊs2࠶ࡿ࠸ࡣ s2ࠊs3ࢆ⏝࠸࡚≀⌮㔞ࢆ j 㠃 ࠿ࡽእᤄࡍࡿࠋ౛࠼ࡤࠊFig. 1 ࡢሙྜࠊ ܽᇱ ௜ǡ௝ିଵൌ൫ݏଵܽ௜ǡ௝൅ ݏଶܽ௜ିଵǡ௝൯Ȁ ሺݏଵ൅ݏଶሻ  ܽൌߩǡ ݌ǡ ݑǡ ݒሺͳͷሻࠋ (c) ( i, j-2 )Ⅼ㸸 ࢩࣥࣉࣝ࡟ j-1 㠃࠿ࡽ≀⌮㔞ࢆእᤄࡍࡿࠋࡍ࡞ࢃࡕࠊ ܽᇱ ௜ǡ௝ିଶൌܽᇱ௜ǡ௝ିଵ         ܽൌߩǡ ݌ǡ ݑǡ ݒ ሺͳ͸ሻࠋ ࡑࡢ௚ࠊ౛እࢣ࣮ࢫ࡜ࡋ࡚ࠊ⩼ࡢᚋ➃㒊ศࡢࡼ࠺࡟ࠊ࢖ ࣥࢱ࣮ࣇ࢙࣮ࢫࡢ㛫㝸ࡀᑠࡉ࠸ࡓࡵࠊ≀య࡟᭱ࡶ㏆࠸᱁Ꮚ Ⅼ⩌࠿ࡽ࡞ࡿ㸯ᒙࡢࡳࡀᏑᅾࡋࠊ㸰ᒙ┠ࡢ᱁ᏊⅬ⩌ࡀᐃ⩏ ࡛ࡁ࡞࠸ሙྜ࡟ࡣࠊ༢⣧࡟࿘ᅖࡢὶయ㒊ศࡢ≀⌮㔞ࢆᖹᆒ ࡋ࡚୚࠼ࡿࠋ౛࠼ࡤࠊFig.2 ࡢࡼ࠺࡞ሙྜࠊ ܽᇱ ௜ǡ௝ൌ൫ܽ௜ାଵǡ௝൅ ܽ௜ǡ௝ାଵ൅ ܽ௜ǡ௝ିଵ൯Ȁ͵ ܽൌߩǡ ݌ǡ ݑǡ ݒ  ሺͳ͹ሻࠋ ᐇ㝿࡟ࡣࠊ≀యෆࡢ㸰ᒙ┠ࡢಟṇ((㸯㸴)ᘧ)ࢆ┬␎ࡋࡓࡾࠊ 㸯ᒙ┠ࡢಟṇࢆ(㸯㸳)ᘧࡢࡼ࠺࡞㔜ࡳ࡙ࡅィ⟬࡛ࡣ࡞ࡃࠊ ┤᥋( i, j )Ⅼ࠿ࡽࡢእᤄ࡛୚࠼࡚ࡶ༑ศ᭷ព࡞⤖ᯝࡀᚓࡽࢀ ࡿࡀࠊୖグࡢಟṇ๎ࢆ⏝࠸ࡓ᪉ࡀⱝᖸࡢ⤖ᯝࡢྥୖࡀぢࡽ ࢀࡓࠋ ௨ୖࡢಟṇࡣࠊࣝࣥࢤࢡࢵࢱἲࡢྛࢫࢸࢵࣉᚋ࡟⾜࠺ࠋ ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫࡢἲ⥺࣋ࢡࢺࣝࡀ⡆༢࡟ᐃ⩏࡛ࡁࡿࣞ࣋ ࣝࢭࢵࢺ㛵ᩘࡢ≉ᛶࢆ⏝࠸ࢀࡤࡇࡢಟṇࡣᐜ࡛᫆࠶ࡾࠊゎ ᯒࢥ࣮ࢻࡢᵓᡂ࡜࠸࠺ほⅬ࠿ࡽࡳࡿ࡜ࠊ▴ᙧ᱁Ꮚࢆ᝿ᐃࡋ ࡓᇶᮏࢥ࣮ࢻ࡟ࠊึᮇ᮲௳࡜ࡋ࡚ࡢࣞ࣋ࣝࢭࢵࢺ㛵ᩘ࡟ࡼ ࡿ≀యࡢᙧ≧ᐃ⩏࡜ࠊ≀⌮㔞ࡢಟṇ࡟㛵ࡍࡿᡭ⥆ࡁࢆᤄධ ࡍࡿࡔࡅ࡛῭ࡴ࡜࠸࠺⡆౽ᛶࢆ᭷ࡋ࡚࠸ࡿࠋ

Fig. 1 Schematic image around interface (Normal condition).

Fig. 2 Schematic image around interface (Special condition). 㸲㸬᳨ドゎᯒ ࡇࡇ࡛ࡣࠊᮏ᪉ἲࡢᛶ⬟ࢆホ౯ࡍࡿࡓࡵࠊ࠸ࡃࡘ࠿ࡢᇶ ᮏⓗ࡞ၥ㢟ࢆ᳨ドࡍࡿࠋ࡞࠾ࠊ࠸ࡎࢀࡢࢣ࣮ࢫ࡟࠾࠸࡚ࡶࠊ 㸦㸯㸲㸧ᘧ୰ࡢࢥࣥࣃࢡࢺࢫ࣮࣒࢟࡟㛵ࡍࡿࣃ࣓࣮ࣛࢱD ࡣ  ࡜ࡋࡓࠋ 㸲㸬㸯 㸰ḟඖ㑄㡢㏿⩼ ࡲࡎࠊ᭱ࡶᇶᮏⓗ࡞᳨ドၥ㢟࡜ࡋ࡚ࠊ୍ᵝὶ࣐ࢵࣁᩘ 0.8ࠊ㏄ゅ 1.25r࡟࠾ࡅࡿ NACA0012 ⩼ᆺ࿘ࡾࡢὶࢀࢆ᳨

ウࡍࡿࠋࡇࡢὶࢀࡣᚑ᮶ࡼࡾ᳨ド౛࡜ࡋ࡚ྲྀࡾୖࡆࡽࢀ࡚ ࠾ࡾ(౛࠼ࡤᩥ⊩(8))ࠊୖ㠃ࡢ⣙ 60%ࢥ࣮ࢻ㛗௜㏆࡟ẚ㍑ⓗ ᙉ࠸⾪ᧁἼࠊୗ㠃ഃ⣙ 35%ࢥ࣮ࢻ㛗௜㏆࡟ᙅ࠸⾪ᧁἼࡀ⏕ ࡌࡿࡇ࡜ࡀࢃ࠿ࡗ࡚࠸ࡿࠋ ᮏ◊✲࡛ࡣࠊ࢖ࣥࢱ࣮ࣇ࢙࣮ࢫࣔࢹࣝࡢᛶ㉁ࡢᢕᥱ࡜࠸ ࠺ᛶ㉁ୖࠊ࡝ࢀ఩ᑡ࡞࠸᱁Ꮚ࡛ୖୗ㠃ࡢ⾪ᧁἼࢆṇࡋ࠸఩ ⨨࡛ᤕࡽ࠼ࡽࢀࡿ࠿࡟╔┠ࡋࠊ㸯ࢥ࣮ࢻ㛗ෆ࡟ࡑࢀࡒࢀ㸳 㸮Ⅼ ('x=0.02) ཬࡧ㸯㸮㸮Ⅼ('x=0.02) ࡢ㸰✀㢮࡛ィ⟬ࢆ⾜ ࡗࡓࠋࡓࡔࡋࠊ᭱኱⩼ཌ࡟࠾ࡅࡿ᱁ᏊⅬࡣ⣙㸰㸮Ⅼ ('y=0.00625) ࡟⤫୍ࡋࡓࠋ࡞࠾ࠊὶ᮰ࡢホ౯࡟ࡣ Roe scheme ࢆ⏝࠸ࠊ᫬㛫้ࡳࡣ CFL=0.6 ࡛࠶ࡿࠋ Fig. 3 ཬࡧ Fig. 4 ࡟ࡑࢀࡒࢀࡢィ⟬᱁Ꮚ࡟ᑐࡍࡿ࣐ࢵࣁᩘ ศᕸࢆ♧ࡍࠋ㸯ࢥ࣮ࢻ㛗ෆ࡟㸳㸮Ⅼ⛬ᗘࢆศᕸࡉࡏࡓ⢒࠸ ᱁Ꮚࢣ࣮ࢫ࡟࠾࠸࡚ࡶ༑ศ࡞⾪ᧁἼゎീᛶ⬟ࢆ᭷ࡍࡿࡇ࡜ ࡀࢃ࠿ࡿࠋ

Fig. 3 Mach Contours (NACA0012, M=0.8, D = 1.25deg.,'x=0.02,'y=0.00625).

Fig. 4 Mach Contours (NACA0012, M=0.8, D = 1.25deg.,'x=0.01,'y=0.00625). 㸲㸬㸰 ㉸㡢㏿ὶ୰ࡢ≀యࢆ㐣ࡂࡿὶࢀ  ḟ࡟ࠊ࣐ࢵࣁᩘ 3.5 ࡢ㉸㡢㏿ὶ࡟ᅛᐃࡉࢀࡓࣉࣜࢬ࣒ ᙧ≧ཬࡧ෇ᰕᙧ≧ࢆ㐣ࡂࡿὶࢀࢆ⪃࠼ࡿࠋୖୗ㠃ࢆቨ㠃࡜ ௬ᐃࡋࡓ Lx=120 mmࠊLy=30 mm ࡢࢲࢡࢺෆ࡟≀యࢆ⨨ࡁࠊ ึᮇ᮲௳ࡣࠊሺߩǡ ݑǡ ݌ሻ ൌ ሺͳǡ͵ǤͷǡͳȀͳǤͶሻ࡜ࡍࡿࠋࣉࣜࢬ࣒ඛ ➃㒊ࡢ༙㡬ゅࡣ 20rࠊ㛗ࡉ 11 mmࠊࡲࡓࠊ෇ᰕ┤ᚄࡣ 5 mm ࡜ࡋࠊ᱁Ꮚᩘࡣࠊ୧⪅࡜ࡶ 1001×251 Ⅼ࡛࠶ࡿࠋὶ᮰ࡢ ホ౯࡟ࡣ LLF(9)_{ࢆ౑⏝ࡋࠊCFL=0.5 ࡜ࡋࡓࠋ} Fig. 5 ཬࡧ Fig. 6 ࡟ࡑࢀࡑࢀࡢᐦᗘศᕸࢆ♧ࡍࠋ⌮ㄽᘧ࠿ ࡽ ⟬ ฟ ࡉ ࢀ ࡿ ࣉ ࣜ ࢬ ࣒ ඛ ➃ ࠿ ࡽ ᨺ ᑕ ࡉ ࢀ ࡿ ⾪ ᧁ Ἴ ゅ ࡣ 35.58r(2)_{࡛࠶ࡿࡢ࡟ᑐࡋࠊFig. 5 ࠿ࡽồࡵࡽࢀࡿ⾪ᧁἼゅ} ࡣ⣙ 35.6rࠊࡲࡓࠊ෇ᰕ๓᪉࡟⏕ࡌࡿ෇ᰕ┤ᚄ D ࡛つ᱁໬ ࡉࢀࡓ㞳⬺⾪ᧁἼ㊥㞳/xࡢ⌮ㄽ್ࡣ/x/D=0.293(2)࡛࠶ࡿࡢ ࡟ᑐࡋࠊFig. 6 ࠿ࡽồࡵࡽࢀࡿ್ࡣࠊ/x/DҸ0. 3 ࡛࠶ࡗࡓࠋ ࡇࢀࡽࡢ᳨ウ࠿ࡽࠊᮏ◊✲࡟࠾ࡅࡿᡭἲࡣࠊ㉸㡢㏿ὶ୰࡟ ࠾ࡅࡿ≀య㏆ഐࡢ⾪ᧁἼᵓ㐀ࢆྵࡴὶࢀሙࢆࠊ࠿࡞ࡾṇ☜ ࡟ᤕࡽ࠼࡚࠸ࡿࡇ࡜ࡀ☜ㄆࡉࢀࡓࠋ 㸲㸬㸱 0RYLQJVKRFNREVWDFOHLQWHUDFWLRQ ᭱ᚋ࡟ࠊ⛣ືࡍࡿ⾪ᧁἼ࡜≀య࡜ࡢ㐃ᡂၥ㢟ࢆ᳨ドࡍࡿࠋ ࡇࡇ࡛ࡣࠊ⛣ື࣐ࢵࣁᩘ Ms=1.30 ࡛ືࡃ⾪ᧁἼࡀࣉࣜࢬ࣒ ᙧ≧ࢆ㏻㐣ࡍࡿ࠸ࢃࡺࡿ Schardin ࡢၥ㢟ࢆྲྀࡾୖࡆࡓ(2)_ࠋ Fig.7 ࡟ᩥ⊩(2)ࡼࡾᘬ⏝ࡋࡓタᐃᅗࢆ♧ࡍࠋࣉࣜࢬ࣒ඛ➃ ࡢ༙㡬ゅࡣ 30rࠊ㧗ࡉ b= 20mmࠊඛ➃㒊ࡢ x ᗙᶆࢆ 54mm ࡜ࡋࠊィ⟬㡿ᇦࡣ Lx=200mmࠊLy=150mm ࡜ࡋࡓࠋࡲ ࡓࠊ⾪ᧁἼࡼࡾྑഃࡢపᅽ㡿ᇦࡢᅽຊࡣ 0.05MPࠊ ᗘࡣ 300K ࡛࠶ࡿࠋධຊ⾪ᧁἼࡢึᮇ఩⨨ࡣࣉࣜࢬ࣒ඛ➃࡜ࡋࠊ ࡑࡇ࠿ࡽࡢ⤒㐣᫬㛫ࢆ t ࡜ࡍࡿࠋィ⟬᱁Ꮚᩘࡣ 2001×1501 Ⅼ,ࠊὶ᮰ࡢホ౯࡟ࡣ Roe scheme ࢆ⏝࠸ࠊ᫬㛫้ࡳࡣ CFL = 0.5 ࡛࠶ࡿࠋ

Fig.8 ࡟⤒㐣᫬㛫 t=138Psec ࡟࠾ࡅࡿࠊshadowgraph ࢖࣓࣮ ࢪࢆ♧ࡍࠋධຊ⾪ᧁἼࡀ≀యࢆ㏻㐣ࡋࡓᚋ࡛ࡣࠊ୍⯡࡟཯ ᑕ⾪ᧁἼࡸ ࠊࡑࡢ௚࡟ࢫࣜࢵࣉࣛ࢖ࣥࠊ࣐ࢵࣁࢫࢸ࣒ࠊ ࡑࢀࡽࡀ஺ᕪࡍࡿ㸱㔜Ⅼ➼ࡢ㠀ᖖ࡟」㞧࡞ὶࢀሙࡀᙧᡂࡉ ࢀࡿࡀࠊ࠸ࡎࢀࡶ㩭᫂࡟ᤕࡽ࠼ࡽࢀ࡚࠾ࡾࠊ఩⨨ࡸᙧ≧ࡶ ᩥ⊩(2)࡟♧ࡉࢀࡓᐇ㦂⤖ᯝ㸦ᩥ⊩(2), Fig.14㸧࡜㠀ᖖ࡟Ⰻࡃ ୍⮴ࡋ࡚࠸ࡿࠋࡲࡓࠊt=28 ࠿ࡽ 178Psec ࡲ࡛ࡢᐦᗘศᕸࡢ ᫬⣔ิኚ໬ࢆ Fig. 9 ࡟♧ࡍࠋ࠸ࡎࢀࡢ⤒㐣᫬㛫࡟࠾࠸࡚ࡶࠊ ὶࢀሙࡢヲ⣽ࢆⰋࡃᤕࡽ࠼࡚࠾ࡾࠊᮏ◊✲࡟࠾ࡅࡿ࢖ࣥࢱ ࣮ࣇ࢙࣮ࢫࡢྲྀࡾᢅ࠸ࡀࡇࡢၥ㢟ࡢࡼ࠺࡞㠀ᐃᖖၥ㢟ࡢヲ ⣽࡞ὶࢀሙࡢゎᯒ࡟ࡶ㐺⏝࡛ࡁࡿࡇ࡜ࡀᐇドࡉࢀࡓࠋ 㸳㸬ࡲ࡜ࡵ 㧗㏿ὶࢆᑐ㇟࡜ࡋࡓᅛẼ┦㐃ᡂၥ㢟࡟ᑗ᮶ⓗ࡟㐺⏝ྍ⬟ ࡛ࠊ࠿ࡘ⡆౽࡞࢖ࣥࢱ࣮ࣇ࢙࣮ࢫࣔࢹࣝࢆᥦ᱌ࡋࡓࠋ≀య ᙧ≧࡟ࡣࣞ࣋ࣝࢭࢵࢺ㛵ᩘࢆ⏝࠸ࠊᮏ◊✲࡛ࡣࠊ≉࡟ࠊⴭ ⪅ࡽ࡟ࡼࡾ᪂ࡋࡃᥦ᱌ࡉࢀࡓᚑ᮶ࡼࡾሀ∼ᛶࡢྥୖࡋࡓ WCNS ࢫ࣮࣒࢟㸦㔜ࡳ௜ࢥࣥࣃࢡࢺࢫ࣮࣒࢟㸧ࢆ⤌ࡳྜࢃ ࡏࡿࡇ࡜࡟ࡼࡾࠊ㧗㏿ὶ୰ࡢ⾪ᧁἼ➼ࡢ୙㐃⥆ࢆྵࡴヲ⣽ ࡞ὶࢀሙࢆ࢜࢖ࣛࣜ࢔ࣥⓗ࡟ᤕࡽ࠼ࡿࡇ࡜ࡀྍ⬟࡛࠶ࡿࡇ ࡜ࢆࠊ㸰ḟඖ㑄㡢㏿⩼ࠊ㉸㡢㏿ὶ୰ࡢ≀య࿘ࡾࡢὶࢀࠊ⛣ ືࡍࡿ⾪ᧁἼ࡜≀య࡜ࡢ㐃ᡂၥ㢟➼ࡢ᳨ドゎᯒ࡟ࡼࡾ☜ㄆ ࡋࡓࠋ ᮏ᪉ἲࡣࠊᙜ↛࡞ࡀࡽ WCNS ࢫ࣮࣒࢟௨እࡢゎᯒࢫ࣮࢟ ࣒࡟ࡼࡿ≀య࿘ࡾࡢᩘ್ゎᯒ࡟ࡶ㐺⏝ྍ⬟࡛࠶ࡿ࡜࡜ࡶ࡟ࠊ ≀యࡢ⛣ືࠊኚᙧࢆక࠺ၥ㢟࡟ࡶᐜ᫆࡟ᣑᙇྍ⬟࡛࠶ࡿࠋ ⌧ᅾࠊࡇࢀࡽࡢㅖㄢ㢟࡟ᑐࡍࡿ᳨ドࢆᐇ᪋୰࡛࠶ࡿࠋ x y 1.5 2 2.5 3 3.5 -0.5 0 0.5 1 cc 1.32 1.2 1.08 0.96 0.84 0.72 0.6 0.48 0.36 0.24 0.12 0 x y 1.5 2 2.5 3 3.5 -0.5 0 0.5 1 cc 1.32 1.2 1.08 0.96 0.84 0.72 0.6 0.48 0.36 0.24 0.12 0

ཧ⪃ᩥ⊩

(1) 㓇஭ᖿኵ⦅ⴭ, “⣮యࡢᩘ್ࢩ࣑࣮ࣗࣞࢩࣙࣥ, ”୸ၿฟ ∧, (2012).

(2) A. Chaudhuri, A. Hadjadj and A. Chinnayya, “On the use of immersed boundary methods for shock/obstacle interactions”, Journal of Computational Physics, 230, (2011), pp.1731-1748.

(3) ఫ㸪㯮, "㔜ࡳ௜ࡁࢥࣥࣃࢡࢺࢫ࣮࣒࢟ࡢሀ∼ᛶ࠾ࡼ ࡧゎീᗘྥୖ࡬ࡢヨࡳ," ➨ 27 ᅇᩘ್ὶయຊᏛࢩ࣏ࣥࢪ ࣒࢘, (2013), C03-2.

(4) 㯮, ఫ, " Anti-diffusion interface sharpening technique ࢆ ᛂ⏝ࡋࡓ㔜ࡳ௜ࢥࣥࣃࢡࢺࢫ࣮࣒࢟࡟࠾ࡅࡿ⌮᝿Ẽయ ୙㐃⥆᥋ゐ㠃ࡢゎീᗘྥୖ࡟ࡘ࠸࡚," ➨ 27 ᅇᩘ್ὶయ ຊᏛࢩ࣏ࣥࢪ࣒࢘, (2013), C03-3.

(5) G. S. Jiang, C. W. Shu, “Efficient implementation of weighted ENO scheme”, Journal of Computational Physics, 126, (1996), pp.202-228.

(6) X. Deng and H. Zhang, “Developing high-order weighted compact nonlinear schemes”, Journal of Computational Physics, 165, (2000), pp. 22-44.

(7) T. Nonomura and K. Fujii, “Robust explicit formulation of weighted compact scheme”, Computers & Fluids, 85, (2013), pp. 8-18.

(8) 㣤 ሯ , ⸨ ஭ , " ༢ ㄪ ᛶ ࢆ ⥔ ᣢ ࡋ ࡓ 㸰 ḟ ⢭ ᗘ Residual Distribution ἲ࡟ࡘ࠸࡚," ➨ 14 ᅇᩘ್ὶయຊᏛࢩ࣏ࣥࢪ ࣒࢘, (2000), E03-3.

(9) E. F. Toro, “Riemann solvers and numerical methods for fluid dynamics: A practical introduction (3rd _edition)”, Springer, (2009).

Fig. 7 Schematic diagram of Schardin’s problem(2)_.

Fig. 8 Numerical shadowgraph image at t=138Psec.

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Fig.5 Density Contours (prism, M=3.5)