ながれ41（2022）281−290281〔原著論文〕

迎角のついたアーチェリー矢側面の境界層流れの安定性解析
電気通信大学大学院

情報理工学研究科

電気通信大学大学院

情報理工学研究科

電気通信大学大学院

情報理工学研究科

東京電機大学大学院

工学研究科

東京電機大学大学院

工学研究科

東京電機大学大学院

工学研究科

東京電機大学大学院

工学研究科

伊 藤 一 希
守
裕 也
宮 嵜
武∗
董
紫 旭
正 藤 範 一岩 津 玲 磨高 橋 直 也Stability analysis of the boundary layer ﬂow around an archeryarrow with angle of attack
Kazuki Ito, The University of Electro-Communications
Hiroya Mamori, The University of Electro-Communications*Takeshi Miyazaki, The University of Electro-Communications
Zixu Dong, Tokyo Denki University
Norikazu Shodo, Tokyo Denki University
Reima Iwatsu, Tokyo Denki University
Naoya Takahashi, Tokyo Denki University∗

E-mail: miyazaki@mce.uec.ac.jp(Received 22 April 2022; revised 30 May 2022; accepted 8 June 2022; published 25 August 2022)

The linear instability of the boundary layer ﬂow formed along an archery arrow with a small angle of attack
γ is investigated based on the parallel ﬂow approximation. The basic ﬂow ﬁeld is computed numericallyat Reynolds numbers Re = 1.5 × 104 and 2.0 × 104 (using the arrow shaft diameter as a representative
length). Two types (bulge and streamlined) of point are considered. When a streamlined point is attached,
the numerically obtained ﬂow is stationary for γ ≤ 0.7◦ and γ ≤ 0.6◦ at Re = 1.5 × 104 and 2.0 × 104 ,respectively. In contrast, stationary ﬂow is found only for γ ≤ 0.3◦ at Re = 1.5 × 104 , if a bulge point is
attached. The basic ﬂow with both points becomes more stable as γ is increased in the range γ ≤ 0.3◦ . In
the case of the streamlined point, new instability modes are found downstream, as γ increases more than
0.5◦ . The result for the bulge point apparently contradicts with the numerical result that the basic ﬂow
becomes unsteady as γ increases more than 0.3◦ , suggesting that the stability analysis should take accountof the inﬂuence of the ﬂow separation behind the bulge point.
(KEY WORDS): archery arrow, angle of attack, boundary layer ﬂow, linear stability analysis, point shape2022

迎角のついたアーチェリー矢側面の境界層流れの安定性解析

282

1

ॹ‫ݴ‬

ຊ‫Ͱڀݚ‬͸ɼ3 ࣍‫ݩ‬௚઀਺஋‫Ͱࢉܭ‬ಘΒΕͨඇ࣠ରশ

ΦϦϯϐοΫͷΞʔνΣϦʔ‫Ͱٕڝ‬͸ 70m ઌͷ௚‫ܘ‬

12.2cm ͷԁΛৗʹࣹൈ͘͜ͱ͕ཁ‫͞ٻ‬ΕΔɽ໼ͷ଎౓͸͓Αͦ 60m/s Ͱɼγϟϑτͷ௚‫ܘ‬Λ୅ද௕͞ʹͨ͠Re ਺͸ 2.0 × 104 ఔ౓Ͱ͋Δɽ໼पΓͷ‫ڥ‬ք૚ྲྀΕ͕ཚྲྀભҠ͢Δͱۭ‫ؾ‬఍߅͸༗ҙʹ૿Ճ͠ɼԖ௚ํ޲ʹతͣΕ͕ੜ͡ΔͨΊɼ‫ٕڝ‬ͷείΞʹେ͖ͳӨ‫ڹ‬Λ‫͢΅ٴ‬ɽඈᠳதͷ໼पΓͷ‫ڥ‬ք૚ྲྀΕͷཚྲྀભҠϝΧχζϜͷ
ղ໌͓Αͼ੍‫ޚ‬͸εϙʔπྲྀମྗֶͷ‫͔఺؍‬Β΋ॏཁྲྀΛ‫ج‬ຊྲྀͱͯ͠ɼฏߦྲྀۙࣅʹ‫͍ͨͮج‬ઢ‫҆ܗ‬ఆੑղੳΛߦ͍ɼܴ֯ͱ᭲‫ܗ‬ঢ়͕‫ڥ‬ք૚ྲྀΕͷ҆ఆੑʹ‫͢΅ٴ‬Ө‫ڹ‬Λղ໌͢Δɽ࣮‫݁ݧ‬ՌͱͷରԠΛٞ࿦͢Δ৔߹ʹ͸ɼۭؒൃల‫ܕ‬ͷઢ‫҆ܗ‬ఆੑղੳ͕ΑΓ๬·͍͠ͱ͞Ε
͍ͯΔɽ͔͠͠ɼඍখͳܴ֯ͷ͍ͭͨ໼ଆ໘ͷ‫ڥ‬ք૚ྲྀΕʹ͍ͭͯͷৄࡉͳ෩ಎ࣮‫݁ݧ‬Ռ͸‫ݟ‬౰ͨΒͳ͍ͷͰɼ͜͜Ͱ͸࣠ରশྲྀΕʹର͢Δཧ࿦త݁Ռ 8) ͱൺֱ͢ΔͨΊʹɼ࣌ؒൃల‫ܕ‬ͷઢ‫҆ܗ‬ఆੑղੳΛߦͬͨɽ

2

Ͱ͋Δɽ

Miyazaki et al.1, 2) ͸ JAXA ͷ࣓ྗࢧ࣋ఱട૷ஔ෇෩ಎʢMSBS ෩ಎʣ࣮‫ ͱݧ‬JISS ʹ͓͚Δඈᠳ࣮‫͔ݧ‬Βɼ‫ྲྀؾ‬ʹର͢Δ໼ͷܴ͕֯ 0.6◦ Ҏ্ͭ͘ͱɼ‫༻ٕڝ‬᭲ʢ௣‫ܕ‬᭲ʣΛ͚ͭͨ໼ͷଆ໘‫ڥ‬ք૚ྲྀΕ͕ཚྲྀભҠ͠ɼ߅ྗ܎
਺ CD ͕‫૿ʹܹٸ‬Ճ͢Δ͜ͱΛࣔͨ͠ɽOrtiz et al.3) ͸ΞʔνΣϦʔ໼ͷγϟϑτ಺ʹՃ଎౓ηϯαΛૠೖ͠ɼඈᠳதͷ໼ͷ CD ஋Λ‫ܭ‬ଌͨ͠ɽߴ଎౓ϏσΦΧϝϥ
ͷө૾͔Βඈᠳதͷ໼ͷܴ֯͸ 0.5◦ ఔ౓ͱධՁ͞Εͨ44͕ɼ௣‫ܕ‬᭲Ͱ͸ 1.6 × 10 ≤ Re ≤ 1.8 × 10 ͷྖҬʹભҠྖҬ͕ଘࡏ͠ɼྲྀઢ‫ܗ‬᭲Ͱ͸ Re = 2.0 × 104 Ͱ΋૚ྲྀঢ়ଶ͕ҡ࣋͞ΕΔ͜ͱ͕ࣔ͞Εͨɽඇ࣠ରশྲྀͷ௚઀਺஋‫ࢉܭ‬ຊઅͰ͸ɼܴ֯ͷ͍ͭͨ໼पΓͷඇ࣠ରশྲྀͷ 3 ࣍‫ݩ‬௚઀਺஋‫ه͍ͯͭʹࢉܭ‬ड़͢Δɽ·ͨɼ਺஋‫݁ࢉܭ‬Ռͱͯ͠ಘΒΕͨඇ࣠ରশྲྀΛ Fourier ‫਺ڃ‬ల։͢Δ͜ͱ
Ͱɼ֤पճํ޲೾਺ʹରԠ͢Δ੒෼͕ Re ਺΍ܴ֯ʹͲͷΑ͏ʹґଘ͢Δ͔ʹ͍ͭͯड़΂Δɽ

2.1

௚઀਺஋‫ࢉܭ‬ख๏౹Β 10) ͸ϕΞʔγϟϑτʢ໼Ӌͳ͠ʣΛա͗Δඇ࣠
ରশྲྀͷ 3 ࣍‫ݩ‬௚઀਺஋‫ࢉܭ‬Λߦͬͨɽγϟϑτ෦෼Λ൒‫ܘ‬Λ aɼ௕͞Λ 258aʢࡉ௕ൺ 129ʣͷԁபͱ͍ͯ͠
Δɽ᭲‫ܗ‬ঢ়ͱͯ͠ྲྀઢ‫ͱܗ‬௣‫ܕ‬ͷ̎छྨΛߟ͑ΔɽͦΕ‫ڥ‬ք૚ྲྀΕͷཚྲྀભҠ‫ߏػ‬Λղ໌͢ΔͨΊʹ͸ɺ·ͧΕͷ᭲‫ܗ‬ঢ়Λਤ 1 ʹࣔ͢ɽྲྀઢ‫ܗ‬᭲͸௣‫ܕ‬᭲ΑΓ΋௕ͣͦͷ҆ఆੑΛௐ΂Δ͜ͱ͕ඞཁͰ͋Δɽԁ౵पΓͷ͘ɼ௣‫ܕ‬᭲ͷ࠷େ௚‫ܘ‬͸γϟϑτ௚‫ܘ‬ΑΓ΋େ͖͍͜ͱ

࣠ରশͳ‫ڥ‬ք૚ྲྀΕͷઢ‫҆ܗ‬ఆੑʹؔ͢Δ‫ڀݚ‬͸਺ଟ͕ಛ௃Ͱ͋Δɽ͘ଘࡏ͠ 4–6) ɼVinod and Govindarajan7) ͸ԁ౵ଆ໘ʹൃ
ୡ͢Δ‫ڥ‬ք૚ྲྀΕͷ̎࣍ෆ҆ఆੑΛௐ΂͍ͯΔɽ͜ΕΒͷ‫ʹڀݚ‬Αͬͯɼԁ౵पΓͷ‫ڥ‬ք૚ྲྀΕ͸ฏ൘‫ڥ‬ք૚ԁ౵࠲ඪ‫( ܥ‬z, r, θ) ͷ Navier-Stokes ํఔࣜͷఆৗղΛࣹӨ๏Ͱ਺஋తʹ‫ٻ‬Ίͨɽ࠲ඪ‫ܥ‬͸໼࣠ํ޲Λ z ɼ൒‫޲ํܘ‬Λ rɼपճํ޲Λ θ ͱͯ͠ɼ᭲ͱγϟϑτͷ‫͗ܧ‬ྲྀΕʹൺ΂ͯΑΓ҆ఆͰ͋Δ͜ͱ͕ࣔ͞Εͨɽ͔͠͠ɼ ໨ͷத৺఺Λ‫͢ͱ఺ݪ‬Δɽ୅਺‫ࢠ֨ܕ‬ੜ੒๏Ͱ O ‫ܕ‬ͷ‫ܭ‬ओͳղੳର৅͕ඇৗʹࡉ௕͍ʢࡉ௕ൺ 105 ʣԁ౵ͷԼྲྀ ࢉ֨ࢠΛ࡞੒͠ɼ֨ࢠ਺͸ 2073×171×82 Ͱɼ࠷খ֨ࢠ

ҬͰ͋ͬͨͨΊɼઌ୺Ͱ͸ Blasius ‫ڥ‬ք૚͕Ծఆ͞Ε͖ͯͨɽ᭲‫ܗ‬ঢ়ͷ҆ఆੑʹର͢ΔӨ‫ڹ‬ΛධՁ͢ΔͨΊʹ͸ɼ᭲‫ܗ‬ঢ়Λ൓ө͢Δ‫ج‬ຊྲྀΛઃఆ͠ͳ͚Ε͹ͳΒͳ͍ɽ௕୩
઒Β 8) ͸࣠ରশ௚઀਺஋‫͔ࢉܭ‬ΒಘͨΞʔνΣϦʔ໼ͷ‫ڥ‬ք૚ྲྀΕʹରͯ͠ɼฏߦྲྀۙࣅʹ‫͍ͨͮج‬ઢ‫҆ܗ‬ఆੑղੳΛߦ͍ɼ໼पΓͷ‫ڥ‬ք૚ྲྀΕ͸पճํ޲೾਺

෯͸ 1.0 × 10−3 Ͱ͋Δɽۭؒࠩ෼ͷਫ਼౓͸ 2 ࣍Ͱ͋Γɼ಺෦఺͸ 2 ࣍ͷத৺ࠩ෼ɼ‫ڥ‬ք఺Ͱ͸ 2 ࣍ͷยଆࠩ෼ۙ
ࣅΛ࢖༻͢Δɽ࣌ؒੵ෼ʹ͸ 3 ஈ 2 Ґͷ௿༰ྔ RungeKutta ๏Λ࢖༻͢ΔɽRe ਺͸Ұ༷ྲྀ଎ U∞ Λ୅ද଎౓ʹɼγϟϑτͷ௚‫ ܘ‬2a Λ୅ද௕ͱͯ͠ɼRe = 2U∞ a/νͱ͢Δɽ͜͜Ͱɼν ͸ྲྀମͷಈ೪ੑ཰Ͱ͋Δɽ

l = 1, 2 ͷϞʔυʹରͯ͠ઢ‫ܗ‬ෆ҆ఆͰ͋Δ͜ͱΛࣔͨ͠ɽ͔͠͠ɼ֧ཚͷۭؒ૿෯཰ͷ໨҆ͱͳΔ N ஋͸ཚྲྀભҠΛ΋ͨΒ͢΄Ͳେ͖͘ͳ͔ͬͨɽྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠৔߹ʹ͍ͭͯɼҏ౻Β 9) ͸ɼ࣠ରশ௚઀਺஋‫ࢉܭ‬(a) ྲྀઢ‫ܗ‬᭲ɽʹΑͬͯಘΒΕͨ‫ڥ‬ք૚ྲྀΕʹܴ֯ͷޮՌΛ໛ٖͨ͠ม
ಈ੒෼ΛॏͶ߹ͤͯ 2 ࣍ෆ҆ఆੑΛௐ΂ͨɽͦͷ݁Ռɼෆ҆ఆੑͷ੒௕཰͕ઌ୺෇ۙͰ༗ҙʹ૿Ճ͠ɼ໼ʹܴ֯(b) ௣‫ܕ‬᭲ɽ͕ͭ͘͜ͱͰ‫ڥ‬ք૚ྲྀΕͷෆ҆ఆԽͱཚྲྀભҠ͕΋ͨΒ͞ΕΔՄೳੑΛࣔͨ͠ɽ͔͠͠ɼΑΓ৴པੑͷߴ͍҆ఆੑղੳΛߦ͏ͨΊʹ͸ɼԾ૝తͳ‫ج‬ຊྲྀͰ͸ͳ࣮͘ࡍʹܴ֯ͷ͍ͭͨ໼पΓͷ‫ڥ‬ք૚ྲྀΕΛ‫ج‬ຊྲྀͱ͠ͳ͚
Ε͹ͳΒͳ͍ɽ

ਤ 1: ᭲‫ܗ‬ঢ়ͷ֓‫ܗ‬ʢ୅ද௕͸γϟϑτ൒‫ܘ‬ʣɽ

伊藤一希・守

2.2

裕也・宮嵜

武・董

ఆৗঢ়ଶͱඇఆৗঢ়ଶ

௚઀਺஋‫ࢉܭ‬͸ Re = 1.5, 2.0 × 104 Ͱɼܴ֯ 0.1◦ ࠁΈͰ࣮ߦ͞Εͨɽແ࣍‫ ؒ࣌ݩ‬t > 500 Ͱ‫͞ࢉܭ‬Εͨ CD஋͕ҰఆͱͳΔ৔߹ΛఆৗྲྀɼCD ஋͕࣌ؒมಈ͢Δ৔߹Λඇఆৗྲྀͱ൑அ͢Δɽ҆ఆੑղੳͷ‫ج‬ຊྲྀͱͯ͠
͸ɼ1000 ≤ t ≤ 1100 ͷ࣌ؒฏ‫ྲྀۉ‬଎৔Λ༻͍Δɽྲྀઢ‫ܗ‬᭲ͷ Re = 1.5 × 104 ͷ৔߹͸ γ ≥ 0.8◦ Ͱඇఆৗྲྀ
ͱͳΓɼRe = 2.0 × 104 ͷ৔߹͸ γ ≥ 0.7◦ Ͱඇఆৗྲྀ
ͱͳͬͨɽҰํͰɼ௣‫ܕ‬᭲ͷ Re = 1.5 × 104 ͷ৔߹͸γ ≥ 0.4◦ ͰඇఆৗྲྀͱͳΓɼRe = 2.0 × 104 ͷ৔߹͸
γ = 0◦ Ͱ͢ͰʹඇఆৗྲྀͱͳΓɼఆৗྲྀ͸ಘΒΕͳ͔ͬͨɽ௣‫ܕ‬᭲Λ૷ணͨ͠৔߹͸ྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠৔߹ʹൺ΂ͯখ͍ܴ֯͞Ͱඇఆৗੑ͕‫ݱ‬ΕΔɽ਺஋‫͍͓ʹࢉܭ‬ͯྲྀΕ͕ඇఆৗԽ͢Δͱ߅ྗ܎਺΋૿େ͢Δɽ໼ଆ໘ͷ‫ڥ‬ք૚ྲྀΕ͕ෆ҆ఆԽͨ͠΋ͷͱߟ͑ΒΕɼܴ֯΍ Re਺ͷ૿Ճ͕ global instability Λ΋ͨΒ͍ͯ͠ΔɽͦͷϝΧχζϜʹഭΔͨΊͷୈҰஈ֊ͱͯ͠ɼຊ‫Ͱڀݚ‬͸ྲྀઢ‫ܗ‬᭲ͰఆৗղΛಘΒΕͨ Re = 1.5 × 104 , 0 ≤ γ ≤ 0.7◦
ͱ Re = 2.0 × 104 , 0 ≤ γ ≤ 0.6◦ ͷ৔߹ʹՃ͑ͯɼ௣‫ܕ‬᭲ͰఆৗղΛಘΒΕͨ Re = 1.5 × 104 , 0 ≤ γ ≤ 0.3◦ ͷ৔߹ʹ͍ͭͯઢ‫҆ܗ‬ఆੑΛௐ΂Δɽ
ਤ 2 ʹྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠৔߹ͷ‫ڥ‬ք૚ྲྀΕͷ࣠ྲྀ
੒෼ Uz Λࣔ͢ɽRe = 1.5 × 104 , γ = 0.5◦ ͷ‫݁ࢉܭ‬Ռ
Ͱ͋ΔɽҰ༷ྲྀ͸ࢴ໘ࠨ͔ͭԞଆ͔Βखલʹ޲͔͏Α͏ʹྲྀΕ͍ͯΔɽ࣠ํ޲଎౓੒෼͸ෲଆʢԞଆʣͰՃ଎

紫旭・正藤範一・岩津玲磨・高橋直也

283͠ɼ֤पճํ޲೾਺ʹରԠ͢Δ੒෼ʹ෼ղ͢Δɽ
 2π
1
(0)
U (r) =U (r, θ)dθ
2π 0
 2π
1
u(m)
Uz (r, θ) cos mθdθz (r) =
π 0

1 2π
u(m)
(r)
=
Ur (r, θ) cos mθdθ
r
π 0

1 2π
(m)
uθ (r) =Uθ (r, θ) sin mθdθ
π 0

(1)
(2)
(3)
(4)

(m)

͜͜Ͱ U (0) ͸पճํ޲೾਺ m = 0 ʹɼuz,r,θ ͸ m = 0ʹରԠ͢Δ੒෼Ͱ͋Δɽz, r ํ޲ͷ sin ੒෼ͱ θ ํ޲ͷcos ੒෼ͷৼ෯͸ O(10−5 ) ҎԼͰ͋Γɼ‫ج‬ຊྲྀʹ͸ରশੑ͕͋Δɽ
·ͣɼྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 ͷ࣠ରশ(0)(m = 0) ੒෼ Uz(0)ʹண໨͢Δɽਤ 3a ʹ Uzͷ൒‫ํܘ‬(0)(0)޲෼෍Λɼਤ 3b ʹ Uz |γ=0 ͱ Uz |γ=0 ͷࠩΛࣔ͢ɽ࣠ํ޲Ґஔ͸ͦΕͧΕ z = 20 ͱ z = 100 Ͱ͋Δɽ‫ڥ‬ք૚
(0)͸Լྲྀଆʢz = 100ʣͰΑΓް͘ͳΔɽUz

͸‫ڥ‬ք૚಺

ͷԁ౵ද໘෇ۙͰՃ଎͞ΕɼԕํͰ‫ݮ‬଎͞ΕΔɽਤ 3b
ʹ z = 20 ʹ͓͚Δ࣠ରশ੒෼ͷࠩ෼Λ͕ࣔ͢ɼ΄΅ܴ֯ͷ 2 ৐ʹൺྫͯ͠૿Ճ͢Δɽैͬͯɼ͜ͷ࣠ํ޲ྲྀͷ࣠ରশ੒෼ͷมԽ͸ɼܴ֯ʹ͍ͭͯ 2 ࣍ͷඇઢ‫ޮܗ‬Ռͱͯ͠‫ݱ‬Εͯ͘Δɽ·ͨɼԼྲྀଆ΁޲͔͏΄Ͳ࣠ରশ଎
౓੒෼ͷ൒‫޲ํܘ‬෼෍͸େ͖͘มԽ͢Δ.

͞ΕɼഎଆʢखલଆʣͰ‫ݮ‬଎͞ΕΔɽܴ֯ͷӨ‫ڹ‬ʢ࣠ର࣍ʹɼྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 ͷ໼पΓশྲྀͱͷͣΕʣ͸Լྲྀଆ΁޲͔͏ʹͭΕͯΑΓ‫ݦ‬ஶͱͷඇ࣠ରশྲྀͷ (m = 1) ੒෼ uz ʹண໨͢Δʢਤ 4aʣɽͳΔɽ·ͨɼܴ͕֯খ͍͞৔߹ɼ࣠ํ޲଎౓੒෼͕୎ӽ

uz ͕ෛͱͳΔ͜ͱ͸ɼ࣠ํ޲ྲྀ͕ θ = π ͷෲଆͰՃ͠ɼ൒‫ܘ‬ɾपճํ޲ͷ଎౓੒෼ΑΓ΋ 2 ܻҎ্େ͖͍ɽ

଎ɼθ = 0 ͷഎଆͰ‫ݮ‬଎͞ΕΔ͜ͱΛࣔ͢ɽͦͷৼ෯͸

(1)

(1)

΄΅ܴ֯ʹൺྫ͢Δ͜ͱ͕෼͔Γɼখܴ͕֯ͭ͘͜ͱͷओཁͳઢ‫ޮܗ‬Ռͱͯ͠‫ݱ‬Ε͍ͯΔɽ࣮ࡍɼ͜ͷ੒෼ͷৼ෯͸ਤ 3b ͷ࣠ରশ੒෼ͷมԽΑΓ΋ 1 ܻେ͖͘ɼฏ൘
‫ڥ‬ք૚ྲྀΕʹ͓͚ΔʮετϦʔΫߏ଄ʯͷ‫ܗ‬੒ʹྨࣅ͢Δ΋ͷͰ͋Γɼ໼पΓͷ‫ڥ‬ք૚ྲྀΕͷ҆ఆੑʹӨ‫ڹ‬Λ‫ٴ‬΅͢ͱߟ͑ΒΕΔɽ
(m)

࣍ʹɼuz
ਤ 2: ࣠ํ޲଎౓੒෼ Uz ͷՄࢹԽɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ = 0.5◦ ͷ‫ڥ‬ք૚ྲྀΕɽ, m = 2, 3 ͷϑʔϦΤ੒෼Λਤ 4b ͱਤ 4c(2)ʹͦΕͧΕࣔ͢ɽuz ͷৼ෯͸΄΅ܴ֯ͷ 2 ৐ʹൺྫͯ͠૿Ճ͓ͯ͠Γɼ2 ࣍ͷඇઢ‫ޮܗ‬Ռͱͯ͠‫ݱ‬Ε͍ͯΔɽ
(1)·ͨɼͦͷৼ෯͸ uz ͷ΋ͷΑΓ΋ 1 ܻখ͍͞ɽಉ༷
(3)ʹɼuz ͸ܴ֯ͷ 3 ࣍ͷඇઢ‫ޮܗ‬Ռͱͯ͠‫ݱ‬ΕΔ΋ͷͱ

2.3ఆৗඇ࣠ରশྲྀͷϑʔϦΤ‫਺ڃ‬ల։ઢ‫҆ܗ‬ఆੑղੳͷ४උͱͯ͠ɼ3 ࣍‫ݩ‬௚઀਺஋‫͔ࢉܭ‬Βಘͨఆৗඇ࣠ରশྲྀ (Uz , Ur , Uθ ) Λ Fourier ‫਺ڃ‬ల։(2)ࢥΘΕ͕ͨɼͦͷৼ෯͸ uz ʹൺ΂ͯͦΕ΄Ͳখ͘͞ͳ͍ͷͰɼ࣍અͷઢ‫҆ܗ‬ఆੑղੳͰ͸͜ͷ੒෼·ͰΛߟྀͯ͠‫ج‬ຊྲྀΛઃఆ͢Δɽ迎角のついたアーチェリー矢側面の境界層流れの安定性解析

284

1
0

0.8

U(0)
z

-0.02
0.6

-0.04
-0.06

uz

(1)

0.4

-0.08

0.2
0

-0.1
-0.12
0

0.2

0.4

0.6

0.8

1
-0.14

r-1
(0)

(a) Uzͷ൒‫޲ํܘ‬෼෍ɿγ ≤ 0.7◦ , z = 20, 100ɽ

-0.16

0

0.1

0.2

0.015
(1)

(a) uz0.5ͷ൒‫޲ํܘ‬෼෍ɿz = 20, γ = 0.1◦ , 0.2◦ , 0.3◦ , 0.4◦ ,

0.005

0.025
0.02

0

0

U(0)
|
z

-U(0)
| =0
z

0.4

0.5◦ , 0.6◦ , 0.7◦ ʢr = 1.1 Ͱ্͔Βʣɽ

0.01-0.005

0.015

(2)

0.01

-0.015

0.005

uz

-0.01

0
0

0.1

0.2

0.3

0.4

0.5

-0.005

r-1
(0)

(0)

-0.01

◦

-0.015(b) Uz |γ=0 ͱ Uz |γ=0 ͷࠩɿz = 20, γ = 0.7◦ , 0.6◦ , 0.5◦ ,
◦

0.3

r-1

◦

◦

0.4 , 0.3 , 0.2 , 0.1 ʢr = 1.1 Ͱ্͔Βʣɽ(0)
ਤ 3: Uz ͷ൒‫޲ํܘ‬෼෍ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re =4

1.5 × 10 ͷ‫ڥ‬ք૚ྲྀΕ.

(2)(b) uz0.4

0.5

0.3 , 0.2◦ , 0.1◦ ʢr = 1.1 Ͱ্͔Βʣɽ

3.1‫ج‬ຊྲྀͷઃఆ

0.015ඇ࣠ରশྲྀ (Uz , Ur , Uθ ) ʹରͯ͠‫ج‬ຊྲྀ (Ūz , Ūr , Ūθ )
(3)

0.01

uz

u(m)
z (r) cos mθ

0.3ͷ൒‫޲ํܘ‬෼෍ɿz = 20, γ = 0.7◦ , 0.6◦ , 0.5◦ , 0.4◦ ,0.02

Ūz = Uz(0) (r) +0.2◦ઢ‫҆ܗ‬ఆੑղੳ

3


0.1

r-1

3

ΛҎԼͷΑ͏ʹઃఆ͢Δɽ

0

0.005

(5)

0

m=1
(0)

Ūθ = Uθ (r) +

3

m=1

(m)

uθ

-0.005(r) sin mθ

(6)
-0.01

ຊ‫ Ͱڀݚ‬m > 3 ͷϑʔϦΤϞʔυΛߟྀ͠ͳ͍ͷ͸ɼ(3)(c) uzΓ೾਺ΛͰ͖Δ͚ͩখͯ͘͞͠ɼ‫ݻ‬༗஋໰୊ͷߦྻͷα◦਺ల։Λଧ੾Δ͜ͱʹΑΔ҆ఆੑղੳ݁Ռ΁ͷӨ‫ڹ‬͸ɼ

m = 0, 1, 2 ·ͰΛߟྀͨ͠৔߹ͱ m = 0, 1, 2, 3 ·ͰΛߟྀͨ͠৔߹ʹಘΒΕΔ࠷େ੒௕཰ͷ༗ޮ਺ࣈ 3 ܻ໨ͷࠩͱͯ͠‫ݱ‬ΕΔɽ·ͨɼฏߦྲྀۙࣅʹ‫͍ͯͮج‬ઢ‫҆ܗ‬

0.1

0.2

0.3

0.4

0.5

r-1

ઢ‫ܗ‬᎟ཚΛपճํ޲ʹϑʔϦΤ‫਺ڃ‬ల։͢Δࡍͷଧ੾ΠζΛখ͘͞ཹΊΔͨΊͰ͋Δɽ‫ج‬ຊྲྀͷϑʔϦΤ‫ڃ‬0ͷ൒‫޲ํܘ‬෼෍ɿz = 20, γ = 0.7◦ , 0.6◦ , 0.5◦ , 0.4◦ ,0.3 , 0.2◦ , 0.1◦ ʢr = 1.1 Ͱ্͔Βʣɽ(m)

ਤ 4: uzͷ൒‫޲ํܘ‬෼෍ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re =1.5 × 104 , z = 20 ͷ‫ڥ‬ք૚ྲྀΕ.

伊藤一希・守

裕也・宮嵜

武・董

紫旭・正藤範一・岩津玲磨・高橋直也

285ఆੑղੳΛߦ͏ͷͰɼܴ֯ͷ͍͍ͭͯͳ͍࣠ରশྲྀͷ৔

Δɽ͜͜Ͱ σ ํ޲ͷ཭ࢄԽʹ͸ Chebyshev બ఺๏Λ࢖߹ͱಉ༷ʹ൒‫޲ํܘ‬ͷ଎౓੒෼͸ Ūr = 0 ͱ͢Δɽ͔͠

༻ͨ͠ɽબ఺਺ N ʹରͯ͠ɼσ ํ޲ͷ఺ͱ Chebyshev

͠ɼ௣‫ܕ‬᭲Λ૷ண͢Δ৔߹ʹ͸ɼ᭲ͱγϟϑτͷ઀߹෦఺ʹ͸ҎԼͷؔ܎͕͋Δɽz ≤ 4 ʹ‫ྖྲྀٯ‬Ҭ͕ଘࡏ͢ΔͨΊɼ͜ͷΑ͏ͳ৔ॴͰ͸nπ
0≤n≤N −1N −1
σ − σ̂σ ∈ [0, ∞) → x ∈ [−1, 1]x=σ + σ̂

xn = cos൒‫޲ํܘ‬ͷ଎౓੒෼Λແࢹ͢Δฏߦྲྀۙࣅ͸ద༻Ͱ͖
8)

ͳ͍ ɽ

3.2

ฏߦྲྀۙࣅʹ‫ͮ͘ج‬ઢ‫ܗ‬ԽҎԼͷ‫ܗ‬ͷઢ‫֧ܗ‬ཚΛ‫ج‬ຊྲྀʹॏͶΔɽ
⎛ ⎞ ⎛
⎞
uz
vz (r, θ)⎜ ⎟ ⎜⎟⎜ur ⎟ ⎜vr (r, θ)⎟ i(αz−ωt)⎜ ⎟=⎜⎟⎜u ⎟ ⎜v (r, θ)⎟ e
θ
θ
⎝ ⎠ ⎝
⎠
p
q(r, θ)(13)(14)͜͜Ͱ͸ σ̂ ͸ σ ํ޲ͷ Chebyshev ఺ͷ෼෍ΛܾΊΔ࣮਺Ͱɼγϟϑτද໘෇ۙͰͷ఺ͷૈີΛௐ੔͢Δɽσ̂ ͕খ͍͞΄Ͳγϟϑτ෇ۙͰ఺͸ີʹ෼෍͢Δɽຊ‫Ͱڀݚ‬(7)͸ɼ௕୩઒Β 8) ʹͳΒͬͯ N = 64ɼσ̂ = 5 ͱͨ͠ɽθ ํ޲ͷ཭ࢄԽʹ͸ࣜ (15) ͷෳૉ Fourier ‫਺ڃ‬ల։Λ͜͜Ͱɼuz , ur , uθ ͱ p ͸ɼͦΕͧΕ଎౓֧ཚͷ֤੒෼

࢖༻ͨ͠ɽ
⎛

⎞
⎞
⎛
vz (r, θ)Al (r)⎟⎟⎜L ⎜
⎜vr (r, θ)⎟⎜ Bl (r) ⎟ ilθ
⎟=
⎟e
⎜
⎜
⎟
⎜v (r, θ)⎟⎜⎠ l=−L ⎝ Cl (r) ⎠⎝ θq(r, θ)Dl (r)

ͱѹྗ֧ཚΛද͢ɽ࣌ؒൃల‫ܕ‬ͷઢ‫҆ܗ‬ఆੑղੳΛߦ͏ͨΊɼ֧ཚͷ࣠ํ޲೾਺Λද͢ӈลͷ α ͸࣮਺ͱ͢Δɽ·ͨ ω ͸ෳૉ֯प೾਺Ͱ͋Γɼͦͷ‫ڏ‬෦ ωi ͸֧ཚͷ੒௕཰Λද͢ɽ΋͠ͻͱͭͰ΋ ωi > 0 ͱͳΔ‫ݻ‬༗஋͕͋(15)Ε͹ɼͦͷ֧ཚ͕੒௕͠ɼ‫ج‬ຊྲྀ͸ઢ‫ܗ‬ෆ҆ఆͱͳΔɽ ҎԼͷઢ‫҆ܗ‬ఆੑղੳͰ͸ɼܴ֯ γ ΍࣠ํ޲࠲ඪ z ʹҰํɼશͯͷ‫ݻ‬༗஋Ͱ ωi < 0 ͳΒ͹ɼ‫ج‬ຊྲྀ͸ઢ‫ ҆ܗ‬Ԡͯ͡ɼల։߲਺Λ 8 ≤ L ≤ 15 ͷྖҬͰมԽͤͨ͞ɽ
ఆͱͳΔɽ࣮෦ ωr ͸֧ཚͷ֯ৼಈ਺Λද͢ɽઢ‫҆ܗ‬ఆੑղੳͰ࢖༻͢Δઢ‫֧ܗ‬ཚํఔࣜ͸ҎԼͰ͋Δɽ͜ͷͱ͖ɼl = L ʹରԠ͢Δ‫ݻ‬༗ؔ਺ͷৼ෯͸ O(10−2 )ҎԼͰ͋Γɼωi ͸গͳ͘ͱ΋ 2 ܻҎ্ͷҰகΛ֬ೝͨ͠ɽҎ্ͷ཭ࢄԽʹΑΓɼࣜ (9)∼(11) ͸ r ํ޲ͷৗඍvθ ∂ ŪzdŪzŪθ ∂vz++
vr + iαq
r ∂θ
r ∂θ
dr
2 ∂ 2 vz
1 ∂ 2 vz
1 ∂vz
=
+
+
− α 2 vz
(8)
Re ∂r2r ∂r
r2 ∂θ22Ūθ
2 ∂ 2 vrŪθ ∂vr∂qi(αŪz − ω)vr +
−
vθ +
=
r ∂θ
r
∂r
Re ∂r22
1 ∂ vr
1 ∂vr
1
2 ∂vθ
+ 2
(9)
+
− α 2 + 2 vr − 2
2
r ∂r
r ∂θ
r
r ∂θ
dŪθ
vθ ∂ ŪθŪθŪθ ∂vθ
i(αŪz − ω)vθ +
+
+
+
vr
r ∂θ
r ∂θ
dr
r
2 ∂ 2 vθ1 ∂ 2 vθ
1 ∂q
1 ∂vθ
=
+ 2
+
+
2
r ∂θ
Re ∂r
r ∂r
r ∂θ2
1
2 ∂vr
(10)
− α 2 + 2 vθ + 2
r
r ∂θ
vr
1 ∂vθ
∂vr
+
+
=0
(11)
iαvz +∂rr
r ∂θ᎟ཚ͕ຬ଍͢Δ‫ڥ‬ք৚݅͸ԁ౵ද໘Ͱͷ೪ண৚݅ͱແ
i(αŪz − ω)vz +‫ݶ‬ԕͰͷ Dirichlet ৚݅Ͱ͋Δɽvz = vr = vθ = 0 (r = 0ɼr → ∞)(12)෼ํఔࣜͱͳΓɼα Λύϥϝʔλͱͨ͠ ω, v ͷ‫ݻ‬༗஋໰୊ʹ‫ؼ‬ண͞ΕΔɽ͞Βʹѹྗ߲ Dl (r) Λফ‫͢ڈ‬Δ͜
ͱͰม਺ͷ਺Λ‫ݮ‬Β͠ɼ3(N − 2)(2L + 1) ࣍ͷ‫ݻ‬༗஋໰୊ͱ͢Δɽ‫ݻ‬༗஋‫ࢉܭ‬͸ MathWorks ࣾͷ਺஋ղੳιϑτ΢ΣΞ MATLAB ͷ eig ؔ਺Λ࢖༻ͨ͠ QZ ๏Ͱߦ͏ɽ֤ z ʹ͓͍ͯɼ࣠ํ޲೾਺ α ͷ༷ʑͳ஋ʹରͯ͠‫ݻ‬༗஋Λ‫͠ࢉܭ‬ɼωi Λ࠷େͱ͢Δ αmax Λ୳͢ɽ4ઢ‫҆ܗ‬ఆੑղੳ݁Ռ
ຊઅͰ͸ɼܴ͕֯খ͘͞‫ڥ‬ք૚ྲྀΕ͕҆ఆԽ͞ΕΔ৔

߹ͱɼܴ͕֯େ͖͘৽͍͠λΠϓͷෆ҆ఆੑ͕‫ݱ‬ΕΔ৔
߹ͱʹ෼͚ͯɼઢ‫҆ܗ‬ఆੑղੳ݁ՌΛ঺հ͢Δɽ

4.1

ܴ͕֯খ͍͞৔߹ʢγ ≤ 0.3◦ ʣɿྲྀઢ‫ܗ‬᭲ɾ௣‫ܕ‬᭲ྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ ≤ 0.3◦ ͷ৔߹ͷ‫࠷ॴہ‬େ੒௕཰ ωi Λਤ 5 ʹࣔ͢ɽԣ࣠͸࣠ํ޲࠲ඪz Λද͢ɽഁઢ͸ܴ͕͍͍֯ͭͯͳ͍৔߹ʢγ = 0◦ ʣͷ݁ՌΛࣔ͢ɽ্ྲྀଆͷෆ҆ఆੑ͸पճํ޲೾਺ l = 2 ͷϞʔυʹରԠ͠ɼԼྲྀଆ͸पճํ޲೾਺ l = 1 ͷϞʔυࣜ (9)∼(11) ͸ภඍ෼ํఔࣜͰ͋ΔͨΊɼ൒‫ͱ޲ํܘ‬पͷ੒௕཰Λද͢ɽਤ 5 ͔Β෼͔ΔΑ͏ʹɼܴ͕֯૿Ճ͢

ճํ޲ͷ 2 ํ޲ʹͦΕͧΕ཭ࢄԽ͠ɼ‫ݻ‬༗஋໰୊ʹ‫ؼ‬ΔʹͭΕͯ ωi ͸୯ௐʹ‫ݮ‬গ͢Δɽपճํ޲೾਺ l = 2

ணͤ͞Δඞཁ͕͋ΔɽͷϞʔυͱ l = 1 ͷϞʔυͷ੾ΓସΘΔ৔ॴ͸ܴ֯ͱͱ

3.3

΋ʹ্ྲྀଆʹҠಈ͢Δ͕ɼͦΕͧΕͷϞʔυͷಛ௃͸΄‫ݻ‬༗஋໰୊ͷఆࣜԽ‫ ݻ‬༗ ஋ ໰ ୊ Λ ਺ ஋ త ʹ ղ ͘ ͨ Ί ʹ ɼζ=2(z/Re)1/2 , σ = (r − 1)/ζ ͷ࠲ඪม‫ ׵‬6) Λಋೖ͢ͱΜͲมΘΒͳ͍ɽ࣮ࡍɼਤ 6 ʹɼ༷ʑͳܴ֯ʹରͯ͠࠷େ੒௕཰Λ༩͑Δ αmax Λ z ͷؔ਺ͱ͕ͯࣔ͢͠ɼܴ

迎角のついたアーチェリー矢側面の境界層流れの安定性解析

286͕͍͍֯ͭͯͳ͍৔߹ʢഁઢʣͱͷྨࣅੑ͕‫ݟ‬ΒΕΔɽ
0.01·ͨɼܴ֯ γ = 0.2◦ ͷ৔߹ʹ͍ͭͯɼz = 50, 150 ʹ͓͚Δ‫ݻ‬༗ؔ਺ͷ z ੒෼ʢvz ʣͷஅ໘෼෍Λɼਤ 7 ͱਤ 80ʹͦΕͧΕࣔ͢ɽz = 50 Ͱ͸पճํ޲೾਺ l = 2 ͷಛ௃͕ɼz = 150 Ͱ͸पճํ޲೾਺ l = 1 ͷಛ௃͕อͨΕ
i

͍ͯΔɽ

-0.01

ܴ͕֯ͭ͘͜ͱͰ‫ڥ‬ք૚ྲྀΕͷ҆ఆੑ͕ߴ·Δͱ͍͏݁Ռ͸গ͠ҙ֎Ͱ͋Δ͕ɼ෇࿥ 1 ʹࣔ͢Α͏ʹɼܴ֯-0.02ͷมԽʹ൐͏‫ج‬ຊྲྀͷ࣠ରশ (m = 0) ੒෼ͷมԽ͕҆ఆԽʹ‫د‬༩͍ͯ͠ΔɽԾʹ‫ج‬ຊྲྀͷ࣠ରশ (m = 0) ੒෼ͷมԽΛߟྀͤͣʹɼm = 1 ͷ੒෼ͷมԽ෼͚ͩΛ

-0.03

0

25

50

75

100

125

150

175

200

z

ॏͶ߹ͤͨ‫ج‬ຊྲྀΛԾ૝తʹઃఆ͢Δͱɼܴ֯ͷ૿Ճͱͱ΋ʹ ωi ͸୯ௐ૿Ճ͢Δ͜ͱ͕֬ೝͰ͖Δɽ͔͠͠ɼ ਤ 5: ‫࠷ॴہ‬େ੒௕཰ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 ×࣠ରশʢm = 0ʣ੒෼ͷมԽ͕ͦͷෆ҆ఆԽΛଧͪফ͠ 104 , γ ≤ 0.3◦ ͷ‫ڥ‬ք૚ྲྀΕɿഁઢ γ = 0◦ ; ࣮ઢ γ =
͍ͯΔɽ

4

◦

ਤ 9 ʹɼྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 2.0×10 , γ ≤ 0.3

0.1◦ , 0.2◦ , 0.3◦ ʢz = 100 Ͱ্͔Βʣɽͷ৔߹ʹ͓͚Δ‫࠷ॴہ‬େ੒௕཰ ωi Λࣔ͢ɽԣ࣠͸ z Λ

3.5ද͠ɼਤதͷഁઢ͸ܴ͕͍͍֯ͭͯͳ͍৔߹ͷ݁ՌͰ
͋Δ ɽRe ਺͕૿Ճ͢Δͱෆ҆ఆੑ͸‫·ڧ‬Δ͕ɼܴ֯

3

ͷ૿Ճͱͱ΋ʹ‫ڥ‬ք૚ྲྀΕ͕҆ఆԽ͞Ε͍ͯ͘͜ͱ͸ɼ

2.5

8)

◦

Re = 1.5 × 10 , γ ≤ 0.3 ͷ৔߹ͱಉ༷Ͱ͋Δ͜ͱ͕෼͔Δɽैͬͯɼྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠໼ଆ໘ͷ‫ڥ‬ք૚ྲྀΕ
͸ Re = 2.0 × 104 , γ ≤ 0.3◦ Ͱ͸૚ྲྀঢ়ଶΛอͭͱࢥΘΕΔɽ௣‫ܕ‬᭲Λ૷ணͨ͠৔߹ʹ͍ͭͯͷ݁ՌΛਤ 10 ʹࣔ͢ɽܴ֯ͷ͍͍ͭͯͳ͍࣠ରশ‫ڥ‬ք૚ྲྀΕͷઢ‫҆ܗ‬ఆੑղੳͰ΋᭲‫ܗ‬ঢ়ͷ‫ݦ‬ஶͳӨ‫ڹ‬͸ͳ͔͕ͬͨ 8) ɼখ͞ͳܴ֯

max

4

= 0°
= 0.1°= 0.2°= 0.3°

2
1.5
1
0.5
0

0

50

100

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Λ͚ͭͯ΋ਤ 5 ͷྲྀઢ‫ܗ‬᭲ͷ৔߹ͱ΄ͱΜͲಉ݁͡ՌzͱͳΔɽҰํɼγ ≥ 0.4◦ ͷྖҬͷ௚઀਺஋‫Ͱࢉܭ‬͸ඇఆਤ 6: ‫࠷ॴہ‬େ੒௕཰ʹରԠ͢Δ αmax ɿྲྀઢ‫ܗ‬᭲Λ૷ৗঢ়ଶ͕ಘΒΕͨɽԼྲྀҬʢz > 5ʣʹ͓͚Δฏߦྲྀۙணͨ͠ Re = 1.5 × 104 , γ ≤ 0.3◦ ͷ‫ڥ‬ք૚ྲྀΕɿഁઢࣅʹ‫ͮ͘ج‬ઢ‫҆ܗ‬ఆੑղੳͰ͸ɼ‫ڥ‬ք૚ྲྀΕͷෆ҆ఆԽ

γ = 0◦ ; ੺࣮ઢ γ = 0.1◦ ; ੨࣮ઢ 0.2◦ ; ྘࣮ઢ 0.3◦ ɽͷஹީΛશ͘ଊ͑Δ͜ͱ͕Ͱ͖ͣɼ͜ͷࣄ࣮Λઆ໌Ͱ͖ͳ͍ɽฏߦྲྀۙࣅʹ‫҆ͮ͘ج‬ఆੑղੳͰ͸ௐ΂Δ͜ͱ͕Ͱ͖ͳ͍௣‫ܕ‬᭲‫୺ޙ‬ͷണ཭ɾ࠶෇ணྖҬͷෆ҆ఆԽ͕ຊ࣭తͳӨ‫ڹ‬Λ‫͍ͯ͠΅ٴ‬ΔΑ͏Ͱ͋Δɽ͜ͷྖҬͷ҆ఆ͢ɽൺֱͷͨΊʹɼRe = 1.5 × 104 , γ = 0◦ ʢࠇഁઢʣ
ͱ Re = 2.0 × 104 , γ = 0◦ ʢ੨ഁઢʣͷ݁ՌΛՃ͍͑ͯΔɽܴ͕֯େ͖͘ͳΔͱɼ༗ҙʹେ͖ͳ੒௕཰Λ࣋ͭෆੑΛௐ΂ΔͨΊʹ͸ɼฏߦྲྀۙࣅʹΑΒͳ͍ TriGlobal҆ఆϞʔυ͕ԼྲྀҬʹ‫ݱ‬Εͯ͘Δɽ͜ͷෆ҆ఆੑ͸ܴ֯ͳ҆ఆੑղੳ͕ඞཁͱͳΔ 11) ɽͷ͍͍ͭͯͳ͍࣠ରশͳ‫ڥ‬ք૚ྲྀΕͰ‫ݟ‬ΒΕͨ l = 1, 2

4.2

ܴ͕֯େ͖͍৔߹ (γ = 0.5◦ , 0.6◦ , 0.7◦ )ɿྲྀઢ‫ܗ‬᭲ྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠৔߹ͷ௚઀਺஋‫Ͱࢉܭ‬͸ɼRe =1.5×104 , γ ≥ 0.8◦ ɼRe = 2.0×104 , γ ≥ 0.7◦ ͰɼͦΕͧΕඇఆৗঢ়ଶͱͳͬͨɽ͜͜Ͱ͸ɼܴ͕֯ΑΓେ͖ͯ͘ఆ
ৗঢ়ଶͱͳΔ৔߹ʢRe = 1.5 × 104 , γ = 0.5◦ , 0.6◦ , 0.7◦
ͱ Re = 2.0 × 104 , γ = 0.5◦ , 0.6◦ ʣʹ͍ͭͯௐ΂Δɽܴ֯ͷ૿Ճͱͱ΋ʹ্ྲྀҬͷ҆ఆੑ͸‫·ڧ‬Δ͕ɼԼྲྀҬʹ͸৽͍͠ෆ҆ఆੑ͕‫ݱ‬ΕΔɽ
ਤ 11 ʹɼRe = 1.5 × 104 , γ ≥ 0.5◦ʢࠇ࣮ઢʣͱ Re =2.0 × 104 , γ ≥ 0.5◦ ʢ੨࣮ઢʣʹ͍ͭͯͷ݁ՌΛࣔ͢ɽԣ࣠͸࣠ํ޲‫ ཭ڑ‬z Ͱɼॎ͕࣠‫࠷ॴہ‬େ੒௕཰ ωi Λදͷෆ҆ఆੑͱ͸ҟͳΔλΠϓͷ΋ͷͰ͋Γɼܴ֯ γ ΍Re ਺͕େ͖͘ͳΔ΄Ͳ੒௕཰͕େ͖͘ͳΓɼͦͷෆ҆ఆྖҬ্͕ྲྀํ޲ʹ޲͔ͬͯ޿͕Δɽ੒௕཰ͷ z ґଘੑΛৄ͘͠ΈΔͱɼܴ͕֯େ͖͘ͳΔʹͭΕͯෳ਺ͷෆ҆ఆϞʔυ͕ൃੜ͢Δ͜ͱ͕෼͔Δɽ
ਤ 12 ʹࣔ͢ αmax ͷ z ґଘੑʹ΋࠷େ੒௕཰ʹରԠ͢Δ αmax ͷ੾ΓସΘΓ͕‫ݟ‬ΒΕΔɽਤதͷࠇഁઢͱ੨
ഁઢ͸ɼͦΕͧΕ Re = 1.5, 2.0 × 104 ͷ࣠ରশ‫ڥ‬ք૚ྲྀΕʹ͓͚Δ l = 1, 2 Ϟʔυͷ αmax Λࣔ͢ɽ৽͍͠ෆ҆ఆϞʔυͷ αmax ͸ࠇ࣮ઢʢRe = 1.5 × 104 ʣͱ੨࣮ઢ
ʢRe = 2.0 × 104 ʣͰද͞Ε͍ͯΔ͕ɺഁઢͱ͸શ͘ҟ

伊藤一希・守

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287

ਤ 7: ‫ݻ‬༗ؔ਺ vz ͷஅ໘෼෍ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ = 0.2◦ , z = 50 ͷ৔߹ɽࠨ͔Β࣮෦ɼ‫ڏ‬෦ɼઈର஋ɽਤ 8: ‫ݻ‬༗ؔ਺ vz ͷஅ໘෼෍ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ = 0.2◦ , z = 150 ͷ৔߹ɽࠨ͔Β࣮෦ɼ‫ڏ‬෦ɼ

0.01

0.01

0

0

i

i

ઈର஋ɽ

-0.01

-0.02

-0.03

-0.02

0

25

50

75

100

125

150

175

200

z

◦

-0.03

0

25

50

75

100

125

150

175

200

z

ਤ 9: ‫࠷ॴہ‬େ੒௕཰ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 2.0 ×
4

-0.01

◦

ਤ 10: ‫࠷ॴہ‬େ੒௕཰ɿ௣‫ܕ‬᭲Λ૷ணͨ͠ Re = 1.5 ×

10 , γ ≤ 0.3 ͷ‫ڥ‬ք૚ྲྀΕɿഁઢ γ = 0 ; ࣮ઢ γ =0.1◦ , 0.2◦ , 0.3◦ ʢz = 100 Ͱ্͔Βʣɽ

104 , γ ≤ 0.3◦ ͷ‫ڥ‬ք૚ྲྀΕɿഁઢ γ = 0◦ ; ࣮ઢ γ =0.1◦ , 0.2◦ , 0.3◦ ʢz = 100 Ͱ্͔Βʣɽ

ͳΔ z ґଘੑ͕‫ݟ‬ΒΕΔɽRe ਺΍ܴ֯ γ ͷ૿Ճʹ൐ͬͷ੾ΓସΘΓ͕ z = 164 Ͱ‫͓͖ͯى‬Γɼਤ 13 ͱਤ 14 ʹ

ͯ αmax ͸૿Ճ͢Δ͕ɼ࠷ԼྲྀҬʢz > 170ʣͰ͸ Re =1.5 × 104 , γ = 0.6◦ ͷ αmax ͱ Re = 2.0 × 104 , γ = 0.5◦͜ͱ͔Β΋֬ೝ͞ΕΔɽz = 200 ʹ͓͚Δෆ҆ఆϞʔυͷ αmax ͱ͕΄΅ॏͳΔɽͷஅ໘෼෍͸ z = 150 ʹ͓͚Δஅ໘෼෍ʹൺ΂ͯΑΓ

4◦Re = 1.5 × 10 , γ = 0.7 ͷ৔߹ʹ͸ɼෆ҆ఆϞʔυࣔ͢ෆ҆ఆϞʔυͷ z ੒෼ͷઈର஋ͷஅ໘෼෍͕ҟͳΔෳࡶԽͯ͠ɼΑΓഎଆͷ൒‫͍͕ͯͬ޿ʹ޲ํܘ‬Δɽਤʹ迎角のついたアーチェリー矢側面の境界層流れの安定性解析

288͸ࣔ͞ͳ͍͕ɼRe = 1.5 × 104 , γ = 0.6◦ , z = 200 Ͱͷ

2.5ෆ҆ఆϞʔυ͸ Re = 1.5 × 104 , γ = 0.7◦ , z = 140 ͷ΋ͷͱྨࣅ͓ͯ͠Γɼܴ֯ͷ૿Ճʹ൐ͬͯԼྲྀҬʹൃੜ͢

2௚઀਺஋‫Ͱࢉܭ‬͸ɼRe = 1.5 × 104 , γ ≥ 0.8◦ ɼRe =2.0 × 104 , γ ≥ 0.7◦ ͷྖҬͰ‫ڥ‬ք૚ྲྀΕ͸ඇఆৗঢ়ଶͱ
ͳΔ͕ɼԼྲྀҬʹ͓͚Δ͜ΕΒͷෆ҆ఆϞʔυ͕ඇఆৗmaxΔෆ҆ఆϞʔυ্͕ྲྀҬʹ޿͕͍ͬͯ͘͜ͱ͕෼͔Δɽ
1.51Խʹؔ༩͍ͯ͠Δͱߟ͑ΒΕΔɽେ͖ܴ֯͘ͷ͍ͭͨྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠໼पΓͷ‫ڥ‬ք૚ྲྀΕͷෆ҆ఆԽͱཚ0.5ྲྀભҠ͸ɼԼྲྀଆ͔Βঃʑʹਐߦ͢Δ͜ͱ͕ࣔࠦ͞Εɼ௣‫ܕ‬᭲Λ૷ணͨ͠৔߹ʹ͓͚ΔཚྲྀભҠϝΧχζϜͱ

0
100

125

150

175

200

z

͸શ͘ҟͳΔΑ͏Ͱ͋Δɽਤ 12: ‫࠷ॴہ‬େ੒௕཰ʹରԠ͢Δ αmax ɿྲྀઢ‫ܗ‬᭲Λ
0.05૷ணͨ͠ Re = 1.5, 2.0 × 104 ͷ‫ڥ‬ք૚ྲྀΕɿࠇഁઢRe = 1.5 × 104 , γ = 0◦ ; ੨ഁઢ Re = 2.0 × 104 , γ = 0◦ ;0.04

ࠇ࣮ઢ Re = 1.5 × 104 , γ = 0.7◦ , 0.6◦ , 0.5◦ ʢ্͔Βʣ;੨࣮ઢ Re = 2.0 × 104 , γ = 0.6◦ , 0.5◦ ʢ্͔Βʣɽ0.02௣‫ܕ‬᭲Λ͚ͭͨ৔߹ɼ௚઀਺஋‫( ࢉܭ‬Re = 1.5 × 104 )

i

0.03

Ͱ͸ γ ≥ 0.4◦ Ͱ‫ڥ‬ք૚ྲྀΕ͕ඇఆৗԽ͍ͯ͠Δɽ·ͨɼ

MSBS ෩ಎ࣮‫Ͱݧ‬΋ 0.4◦ ≤ γ ≤ 0.6◦ Ͱ߅ྗ܎਺ͷ‫૿ٸ‬

0.01͕ࣔ͞Ε͍ͯΔɽ͜ΕΒ͸௣‫ܕ‬᭲‫ൃʹ୺ޙ‬ੜ͢Δണ཭๐
0
100

125

150

175

200

ʹΑΔ‫ྲྀٯ‬Ҭͷෆ҆ఆԽʹ൐͏΋ͷͰɼฏߦྲྀۙࣅʹ‫ج‬zͮ҆͘ఆੑղੳͰ͸ଊ͑Δ͜ͱ͕Ͱ͖ͳ͍ɽ௣‫ܕ‬᭲Λ૷ਤ 11: ‫࠷ॴہ‬େ੒௕཰ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re =ணͨ͠໼ͷ‫ڥ‬ք૚ྲྀΕͷཚྲྀભҠ‫ݱ‬৅Λղ໌͢ΔͨΊ4◦

1.5, 2.0 × 10 , γ ≥ 0.5 ͷ‫ڥ‬ք૚ྲྀΕɿࠇഁઢ Re =1.5 × 104 , γ = 0◦ ; ੨ഁઢ Re = 2.0 × 104 , γ = 0◦ ; ࠇ࣮
ઢ Re = 1.5 × 104 , γ = 0.7◦ , 0.6◦ , 0.5◦ʢz = 200 Ͱ্͔ʹ͸ TriGlobal ͳ҆ఆੑղੳΛ࣮ࢪ͢Δඞཁ͕͋Δɽ
Ұํɼྲྀઢ‫ܗ‬᭲Λ͚ͭͨ৔߹ʹ͸ɼγ ≥ 0.5◦ ͰԼྲྀҬʹ৽͍͠ෆ҆ఆϞʔυ͕ग़‫ͯ͠ݱ‬ɼͦͷ੒௕཰͸ γ = 0◦

Βʣ; ੨࣮ઢ Re = 2.0 × 104 , γ = 0.6◦ , 0.5◦ʢz = 200 ͰͰͷෆ҆ఆੑͷ੒௕཰ΑΓ΋༗ҙʹେ͖͘ɼRe ਺΍্ܴ͔Βʣɽ֯ γ ͱͱ΋ʹ૿େͨ͠ɽRe ਺΍ γ ͷ૿Ճͱͱ΋ʹɼ͜ͷෆ҆ఆੑΛ‫ܦ‬༝ͨ͠‫ڥ‬ք૚ྲྀΕͷཚྲྀભҠ͕ԼྲྀҬ5͔Β্ྲྀҬʹ޲͚ͯਐΈɼ߅ྗ܎਺Λঃʑʹ૿Ճͤ͞Δ݁‫ݴ‬ͱߟ͑ΒΕΔɽ࣮ࡍɼྲྀઢ‫ܗ‬᭲Λ͚ͭͨΫϩεϘ΢໼ʹຊ‫Ͱڀݚ‬͸ɼܴ֯ͷ͍ͭͨΞʔνΣϦʔ໼ଆ໘ͷ‫ڥ‬ք͍ͭͯͷ MSBS ෩ಎ࣮‫Ͱݧ‬͸ɼ߅ྗ܎਺͕ܴ֯ γ ͱͱ૚ྲྀΕʹରͯ͠ɼฏߦྲྀۙࣅʹ‫ͮ͘ج‬ઢ‫҆ܗ‬ఆੑղੳΛ΋ʹ؇΍͔ʹ૿Ճ͢Δଌఆ݁Ռ͕ࣔ͞Ε͍ͯΔ 12) ɽ·ߦͬͨɽ3 ࣍‫ݩ‬ͷ௚઀਺஋‫͔ࢉܭ‬ΒಘΒΕͨඇ࣠ରশྲྀ
Λपճํ޲ʹ Fourier ‫਺ڃ‬ల։ͯ͠ɼ೾਺ m = 0, 1, 2, 3ͨɼඈᠳ࣮‫ʹݧ‬ΑΔ‫ܭ‬ଌ (Re = 2.0 × 104 , γ ≤ 0.5◦ ) Ͱɼྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠໼ͷ߅ྗ܎਺͕૚ྲྀ஋ͱͳΓɼཚྲྀͷ੒෼·ͰͰଧͪ੾ͬͯ‫ج‬ຊྲྀͱͨ͠ɽ໼ͷઌ୺ʹ͸ભҠ͕‫ͱͱ͍͜ͳ͍͖ͯى‬΋੔߹͢Δɽ܏͍ͨ໼ଆ໘ͷྲྀઢ‫ܗ‬᭲ͱ௣‫ܕ‬᭲Λ૷ணͯ͠ɼRe = 1.5 × 104 ͓Αͼ‫ڥ‬ք૚ྲྀΕͷཚྲྀભҠ‫ݱ‬৅͸᭲‫ܗ‬ঢ়ʹΑͬͯશ͘ҟͳΔ

2.0 × 104 Ͱܴ֯ γ Λ 0.1◦ ࠁΈͰ૿Ճͤͯ͞ఆৗྲྀͱͳ

͜ͱ͕෼͔ΔɽΔྖҬΛௐ΂ͨɽઢ‫֧ܗ‬ཚํఔࣜͷ཭ࢄԽ͸൒‫ʹ޲ํܘ‬෇࿥ 1

Chebyshev બ఺๏ɼपճํ޲ʹ͸ෳૉ Fourier ‫਺ڃ‬ల։Λ࢖༻ͨ͠ɽ
ܴ֯ γ ͕খ͍͞৔߹ (γ ≤ 0.3◦ ) ʹ͸ɼ᭲‫ܗ‬ঢ়΍ Re ਺ʹ͔͔ΘΒͣɼܴ֯ γ ͕૿Ճ͢ΔʹͭΕͯ੒௕཰͸‫ݮ‬গͨ͠ɽ͜Ε͸ɼ‫ج‬ຊྲྀͷ࣠ରশ (m = 0) ੒෼͕ܴ֯ͱ
ͱ΋ʹมԽ͠ɼγ = 0◦ ʹ͓͚Δपճํ޲೾਺ l = 1 ͱl = 2 ͷෆ҆ఆੑ͕҆ఆԽ͞ΕͨͨΊͰ͋Δɽ͔͠͠ɼ

‫ج‬ຊྲྀͷ࣠ରশ੒෼ͷΈʹର͢Δ҆ఆੑղੳ݁Ռ

ຊ෇࿥Ͱ͸ܴ͕֯খ͍͞৔߹ʹ‫ڥ‬ք૚ྲྀΕ͕҆ఆԽͨ͠ཧ༝ʹ͍ͭͯड़΂Δɽਤ 15 ʹྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠৔߹ͷRe = 1.5 × 104 ͷ࣠ରশ੒෼ m = 0 ΛԾ૝తͳ‫ج‬ຊྲྀͱͯ͠ઢ‫҆ܗ‬ఆੑղੳΛߦͬͨ݁ՌΛࣔ͢ɽഁઢ͸্͔Β

γ = 0◦ , 0.1◦ , 0.2◦ , 0.3◦ ͷ৔߹ͷ࠷େ੒௕཰ΛͦΕͧΕද͢ɽ্ྲྀଆ͸पճํ޲೾਺ l = 2 ͷϞʔυɼԼྲྀଆ͸पճํ޲೾

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ਤ 13: ‫ݻ‬༗ؔ਺ vz ͷՄࢹԽɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ = 0.7◦ , z = 140 ͷ‫ڥ‬ք૚ྲྀΕɽࠨ͔Β࣮෦ɼ‫ڏ‬෦ɼઈର஋ɽ

ਤ 14: ‫ݻ‬༗ؔ਺ vz ͷՄࢹԽɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 × 104 , γ = 0.7◦ , z = 200 ͷ‫ڥ‬ք૚ྲྀΕɽࠨ͔Β࣮෦ɼ‫ڏ‬෦ɼઈର஋ɽ
਺ l = 1 ͷϞʔυʹରԠ͢Δɽܴ֯ͷ૿Ճͱͱ΋ʹෆ҆ఆ

0.01

ੑͷ੒௕཰͕‫ݦ‬ஶʹ‫ݮ‬গ͍ͯ͠Δɽ௚઀਺஋‫Ͱࢉܭ‬ಘΒΕͨm = 1, 2, 3 ͷϑʔϦΤ੒෼ΛॏͶ߹ͤͨ‫ج‬ຊྲྀʹର͢Δ݁Ռ
ʢਤ 5ʣͱൺֱ͢Δͱɼ࠷େ੒௕཰ͷ‫ݮ‬গ෯͕େ͖͘ͳ͍ͬͯ

0पΓͷ‫ڥ‬ք૚ྲྀΕ͕҆ఆԽ͞ΕΔओͳ‫ݪ‬Ҽ͸ɼ‫ج‬ຊྲྀͷ࣠ରশ੒෼ͷ൒‫޲ํܘ‬෼෍͕มΘΔ͜ͱʹ͋ΔɽҾ༻จ‫ݙ‬
1) Miyazaki, T., Mukaiyama, K., Komori, Y., Okawa, Y.,

Taguchi, S., and Sugiura, H.: Aerodynamic propertiesof an archery arrow, Sports Engineering, 16 (2013) 4354.
2) Miyazaki, T., Matsumoto, T., Ando, R., Ortiz, J., andSugiura, H.: Indeterminacy of drag exerted on an arrow in free ﬂight: arrow attitude and laminar-turbulent
transition, Eur. J. Phys., 38: 064001 (2017).
3) Ortiz, J., Hasegawa, T., Serino, A., Miyazaki, T., andSugiura, H.: Instability and transition of a boundary
layer ﬂow along an arrow: Relaminarization in freeﬂight tests, XXV ICTAM, 23-28 August 2020, Milano, Italy (2021).

iΔ͜ͱ͕෼͔Δɽैͬͯɼখ͞ͳܴ͕֯ͭ͘͜ͱʹΑͬͯ໼
-0.01

-0.02

-0.03

0

25

50

75

100

125

150

175

200

z

ਤ 15: ‫࠷ॴہ‬େ੒௕཰ɿྲྀઢ‫ܗ‬᭲Λ૷ணͨ͠ Re = 1.5 ×

104 ͷ‫ڥ‬ք૚ྲྀΕͷ࣠ରশ੒෼ΛԾ૝తͳ‫ج‬ຊྲྀͱͨ͠
৔߹ɿγ = 0◦ , 0.1◦ , 0.2◦ , 0.3◦ ʢz = 200 Ͱ্͔Βʣɽ4) Tutty, O. R., Price, W. G., and Parsons, A. T.: Boundary layer ﬂow on a long thin cylinder, Phys. Fluids, 14(2002) 628-637.
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