2023_spring_asj

Transcript

1. վྑ൛Balsara SwitchΛ༻͍ͨɺSPH๏ͷۭ࣋ͭؒθϩ ࣍ޡ͕ࠩ༠ൃ͢ΔِKelvin-Helmholtzෆ҆ఆͷ཈੍ ౬ઙ୓޺, ৿ਖ਼෉ ஜ೾େֶ ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 1 / 13
2. ໨࣍ 1 ֤छεΩʔϜʹ͍ͭͯ 2 ِ Kelvin-Helmholtz ෆ҆ఆ 3 ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 2 / 13
3. ֤छεΩʔϜʹ͍ͭͯ SPH ๏ಋग़ʹ༻͍Δѹॖੑඇ೪ੑஅ೤ྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ dv(r) dt = − 1 ρ(r) ∇P(r) + FAV (α) (1) du(r) dt = − P(r) ρ(r) ∇ · v(r) + GAV (α) (2) • σϝϦοτ ▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺඇ෺ཧతͳද໘ுྗ͕ൃੜ. ີ౓ͷۭؒ࿈ଓੑΛԾఆ͠ɺ࿈ଓؔ਺Ͱۙࣅ͍ͯ͠Δ͔Βɻ ▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁɻ ਓ޻೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ SSPH DISPH GSPH(ࣗಈతʹద੾ͳ೪ੑ) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 3 / 13
4. ֤छεΩʔϜʹ͍ͭͯ Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ (࿦จ౤ߘ༧ఆ) mi dvi dt = − N ∑ j [ fgrad i P∗ ij Ui Uj q2 i ∇i Wij (hi ) + fgrad j P∗ ij Ui Uj q2 j ∇i Wij (hj ) ] (3) dUi dt = fgrad i N ∑ j P∗ ij Ui Uj q2 i vij · ∇i Wij (hi ). (4) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 4 / 13
5. ِ Kelvin-Helmholtz ෆ҆ఆ ۭؒθϩ࣍ޡࠩ F(r) = ∫ F(r′ )W(|r − r′ |, h)d3r′ + O(h2) = ∑ j mj ρj FjW(|r − rj|, h) + O(1) + O(h2) (5) • ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ͢Δ͜ͱ͕஌ΒΕͯ ͍Δ ▶ ཻࢠ෼෍ͷඍົͳҧ͍Ͱ΋஋͕มԽ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 5 / 13
6. ِ Kelvin-Helmholtz ෆ҆ఆ • ਓ޻೪ੑ߲͸ຊདྷিܸ೾ྖҬͰͷΈಇ͍ͯ΄͍͕͠ɺγΞྖҬͰ΋ಇ͍ͯ͠·͏ɻ ▶ rij ͷϕΫτϧ্Ͱཻࢠ͕ޓ͍ʹ͍͍ۙͮͯΔ͔͕িܸ೾ͷ൑ఆ๏ʹͳ͍ͬͯΔ͔Β Balsara switch FAV (α) → 0.5(Fi + Fj)FAV (α), GAV (α) → 0.5(Fi + Fj)GAV (α)ɻ γΞྖҬͰ͸ Fi, Fj Λ 1 ΑΓখ͘͞ɺিܸ೾ྖҬͰ͸ Fi, Fj ͕ 1 ʹͳΔΑ͏ʹઃఆɻ Fi = |∇i · vi| |∇i · vi| + |∇i × vi| + 0.0001ci/hi (6) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 6 / 13
7. ِ Kelvin-Helmholtz ෆ҆ఆ GDISPH ΁ͷ Balsara switch ͷద༻ • GSPH,GDISPH ͸ rij ͷϕΫτϧ্Ͱ 1 ࣍ݩ Riemann solver Λղ͘ɻೖྗ஋Λཻࢠ i ͱ j ͷ෺ཧྔͱ͠ɺग़ྗ஋Λ௨ͯ͠೪ੑ͕ೖΔ ▶ γΞྖҬͱিܸ೾ྖҬͷ۠ผ͸͔ͭͳ͍ɻ • DISPH ͸ (ඇਓ޻೪ੑ߲)+(ਓ޻೪ੑ߲) ʹཅʹ෼͔Ε͍ͯΔ͕ɺGDISPH ͸ 1 ͭͷ߲ʹ ·ͱ·͍ͬͯͯ෼཭͍ͯ͠ͳ͍ɻ • ཧ૝͸ GDISPH தͷ P∗ ij Λ (ඇ೪ੑ߲)+(೪ੑ߲) ʹཅʹ෼͚Δ͜ͱ ▶ ·ͩ͏·͍͍ͬͯ͘ͳ͍ɻBalsara switch ͷద༻ྫ΋ͳ͍ɻ ࣮ݧతʹࢼͯ͠ΈΔ • (DISPH ͷඇਓ޻೪ੑ߲)+0.5(Fi + Fj)(GDISPH ͷ߲ ʔ DISPH ͷඇਓ޻೪ੑ߲) Λɺ GDISPH ͷࣜͱͯ͠༻͍Δɻ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 7 / 13
8. ِ Kelvin-Helmholtz ෆ҆ఆ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 8 / 13
9. ِ Kelvin-Helmholtz ෆ҆ఆ ݪҼͷ༧૝ ϊΠζͷਖ਼ମ ཻ֤ࢠʹ͓͚Δɺۭؒ 0 ࣍ͷޡࠩ΍ O(h2) ͷޡࠩͷ͔͔Γํͷҧ͍ʹΑΔ΋ ͷɻཻ֤ࢠؒͷඍົͳ਺஋ͷͣΕ͕ɺ ඇৗʹ೾௕ͷখ͍͞ઁಈͱͳΔɻ ϊΠζ͕େ͖͘੒௕͢Δཧ༝ • Balsara swtich ͷಋೖʹΑͬͯγΞྖ ҬͰͷ೪ੑ͕΄΅ 0 ʹͳΔɻ • ڪΒ͘෺ཧతʹ͸ɺྲྀମͷ೪ੑ͕େ ͖͘ͳΔͱɺKH ෆ҆ఆੑ͕ى͜Δ ࠷খͷ೾௕΋େ͖͘ͳΔɻ ೪ੑ͕ 0 ͳΒ͹ɺ ͲͷΑ͏ͳ೾௕ʹରͯ͠΋ෆ҆ఆੑ੒௕ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 9 / 13
10. ِ Kelvin-Helmholtz ෆ҆ఆ վળࡦ • ͲͷΑ͏ͳঢ়گʹ͓͍ͯ΋ɺ෺ཧతͳઁಈʹΑΔෆ҆ఆੑͷ੒௕͸ڐ͠ɺ਺஋తͳϊΠ ζʹΑΔઁಈͷ੒௕͸཈͑Δɻ → Balsara swtich ͷڧ͞Λௐઅ͢Δ͜ͱͰ࣮ݱΛ໨ࢦ͢ (γΞʹ͋͑ͯ೪ੑΛՃ͑Δ)ɻ Balsara switch • FAV (α) → 0.5(Fi + Fj)FAV (α), GAV (α) → 0.5(Fi + Fj)GAV (α)ɻ Fi = |∇i · vi| |∇i · vi| + |∇i × vi| + 0.0001ci/hi (7) զʑͷఏҊख๏ • Fi, Fj ͷ୅ΘΓʹ F′ i , F′ j Λ༻͍Δɻ • β ͸ [0, 1] ͷ࣮਺ΛऔΔ೚ҙύϥϝʔλɻβ = 1 Ͱ Balsara swtich Φϯɹ β = 0 Ͱ Balsara swtich Φϑ F′ i = 1 + β(Fi − 1) (8) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 10 / 13
11. ِ Kelvin-Helmholtz ෆ҆ఆ β = 0.5 ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 11 / 13
12. ِ Kelvin-Helmholtz ෆ҆ఆ β = 0.85 ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 12 / 13
13. ݁࿦ ݁࿦ • Kelvin-Helmholtz ෆ҆ఆੑ ▶ ϊΠζʹΑΔઁಈ΋੒௕͢Δ. ղܾࡦͱͯ͠ɺ͋͑ͯγΞྖҬʹ೪ੑΛՃ͑Δ͜ͱΛఏҊ. ํ޲ੑ͸ਖ਼ͦ͠͏. ΍Γํʹؔͯ͠͸վળͷ༨஍͋Γɻ ▶ ͲͷΑ͏ͳঢ়گͰ΋ɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͑ΒΕΔํ๏Λݟ͚͍ͭͨɻ ࠓޙͷ՝୊ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
14. ݁࿦ ग़య Figure: ”Fancy SPH convolution scheme (verbose, modified colors scheme)” created by Jlcercos is licenced under CC BY-SA 4.0(https://creativecommons.org/licenses/by-sa/4.0/) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
15. ݁࿦ Reference I Balsara, D. S. (1995). Von neumann stability analysis of smoothed particle hydrodynamicsŠsuggestions for optimal algorithms. Journal of Computational Physics, 121(2):357–372. Brookshaw, L. (1985). A method of calculating radiative heat diffusion in particle simulations. Publications of the Astronomical Society of Australia, 6(2):207 r 210. Cha, S.-H. and Whitworth, A. P. (2003). Implementations and tests of godunov-type particle hydrodynamics. Monthly Notices of the Royal Astronomical Society, 340(1):73–90. ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
16. ݁࿦ Reference II Garc´ ıa-Senz, D., Cabez´ on, R. M., and Escart´ ın, J. A. (2012). Improving smoothed particle hydrodynamics with an integral approach to calculating gradients. Astronomy & astrophysics, 538:A9. Inutsuka, S.-i. (2002). Reformulation of smoothed particle hydrodynamics with riemann solver. Journal of Computational Physics, 179. Price, D. J. (2008). Modelling discontinuities and kelvin–helmholtz instabilities in sph. Journal of Computational Physics, 227(24):10040–10057. ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
17. ݁࿦ Reference III Saitoh, T. R. and Makino, J. (2013). A DENSITY-INDEPENDENT FORMULATION OF SMOOTHED PARTICLE HYDRODYNAMICS. The Astrophysical Journal, 768(1):44. Wadsley, J. W., Keller, B. W., and Quinn, T. R. (2017). Gasoline2: a modern smoothed particle hydrodynamics code. Monthly Notices of the Royal Astronomical Society, 471(2):2357–2369. ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
18. ݁࿦ DISPH ๏ɺGSPH ๏ͱ͸ • [Saitoh and Makino, 2013] Ͱ DISPH ๏͕ɺ[Inutsuka, 2002] Ͱ GSPH ๏͕։ൃ͞Εͨɻ DISPH ๏ ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛ࿈ଓؔ਺Ͱ ۙࣅ (ѹྗͷ࿈ଓੑɺۭؒඍ෼ՄೳੑΛԾ ఆ) ͠ɺྲྀମͷඍ෼ํఔࣜΛղ͍͍ͯΔɻ িܸ೾Λଊ͑ΔͨΊʹɺਓ޻೪ੑύϥ ϝʔλ͕ඞཁɻ GSPH ๏ ཻࢠಉ࢜ͷ૬ޓ࡞༻ͷܭࢉͷࡍɺܭࢉࣜ தͷѹྗͱ଎౓ʹ Riemann solver Λ༻͍ Δɻࠓճ͸ [Cha and Whitworth, 2003] ʹ ΑΔ Case3 ͷ GSPH Λ࢖༻. িܸ೾ͷͨ Ίͷύϥϝʔλઃఆඞཁͳ͠. ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
19. ݁࿦ Godunov DISPH ๏ʹ͍ͭͯ (࿦จ౤ߘ༧ఆ) [Inutsuka, 2002] ͷํ๏ͱ͸ҧͬͨ, զʑͷΦϦδφϧͷํ๏Ͱ Riemann solver Λ DISPH ʹ૊ ΈࠐΉ Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ ΤωϧΪʔํఔࣜ͸ dUi dt = fgrad i N ∑ j P∗ ij UiUj q2 i vij · ∇iWij(hi). (9) ӡಈํఔࣜ͸ɺ࡞༻൓࡞༻Λຬͨ͠ΤωϧΪʔอଘ΋ຬͨ͞ͳ͚Ε͹ͳΒͳ͍ͱ͍͏৚͔݅ ΒٻΊΒΕΔɻ mi dvi dt = − N ∑ j [ fgrad i P∗ ij UiUj q2 i ∇iWij(hi) + fgrad j P∗ ij UiUj q2 j ∇iWij(hj) ] (10) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
20. ݁࿦ ઀৮ෆ࿈ଓ໘ • ີ౓͸Χʔωϧิ׬ʹΑͬͯٻΊΒΕΔ ρ = ∑ j mjWij(h) (11) • ಺෦ΤωϧΪʔ͸࣌ؒੵ෼ͰٻΊΒΕΔ • ઀৮ෆ࿈ଓ໘Ͱɺີ౓͸ͳΊΒ͔ɺ಺෦ ΤωϧΪʔ͸ٸܹʹมԽ P = (γ − 1)ρu (12) ▶ P ʹෆ੔߹͕ੜ͡ɺͦͷ݁Ռද໘ுྗ͕ ൃੜ [Price, 2008] ▶ ཻࢠͷମੵཁૉ dV = m ρ ͸ີ౓ෆ࿈ଓ ͷͱ͜ΖͰਫ਼౓௿Լ [Saitoh and Makino, 2013] ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
21. ݁࿦ • ߋʹɺಋग़தʹີ౓ͷඍ෼Λ࢖༻ − ∇P ρ = −∇ ( P ρ ) − P ρ2 ∇ρ (13) ෺ཧతʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘Ͱ͸ɺີ౓ͷۭؒඍ෼Λܦ༝ͯ͠ಋग़͞Εͨࣜ ͸ਖ਼͘͠ͳ͍͸ͣ [Saitoh and Makino, 2013] ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
22. ݁࿦ 2D Kelvin-Helmholtz ෆ҆ఆੑ Ґஔ y = 0.25, y = 0.75 ͷͦΕͧΕʹɺy ࣠ํ޲ͷ଎౓ʹ 2 ೾௕෼ͷઁಈΛ༩͍͑ͯΔ ෆ҆ఆੑͷ੒௕ͷλΠϜεέʔϧ τKH = 1.06 ͷ໿ 4 ഒͷ t = 4.0 ·Ͱܭࢉ Balsara switch Λ෇͚ͯܭࢉ ॳظ৚݅ ༩͑ͨઁಈ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
23. ݁࿦ վળࡦ 2 • γΞʹ೪ੑͰ͸ͳ͘ΤωϧΪʔ֦ࢄ͕ಇ͘Α͏ʹͯ͠ΈΔɻࠓճ͸ਓ޻೤఻ಋ߲ΛՃ ͑Δɻ • [Price, 2008] ͕઀৮ෆ࿈ଓ໘Ͱͷඇ෺ཧతͳද໘ுྗΛ཈͑Δख๏ͱͯ͠ߟҊ • ਪ঑஋ͷ αu = 1 Λ࢖༻ SSPH with ArtCond ಋग़ʹ༻͍ΔѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ dv(r) dt = − 1 ρ(r) ∇P(r) + FAV (αAV ) (14) du(r) dt = − P(r) ρ(r) ∇ · v(r) + GAV (αAV ) + HAC(αu) (15) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
24. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
25. ݁࿦ ೪ੑ͕ྑͦ͞͏ͳཧ༝ • ΤωϧΪʔ֦ࢄ߲͸ີ౓͕ಉ͡ʹͳΔΑ͏ಇ͖ɺ೪ੑ߲͸଎౓͕ಉ͡ʹͳΔΑ͏ʹಇ ͘ɻγΞྖҬͰ͸ͳΔ΂͘ಇ͔ͳ͍Α͏ʹ͍ͨ͠ɻ ▶ গͳ͍֦ࢄ or ೪ੑͰɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͍͑ͨ ▶ ΤωϧΪʔ֦ࢄ߲෇͖ or ೪ੑ߲෇͖ͷྲྀମํఔࣜͷઢܗղੳΛߦ͏͜ͱͰௐ΂͍ͨɻࠓޙ ͷ՝୊ • KH ෆ҆ఆੑͷλΠϜεέʔϧΛݟΔͱɺີ౓ͷۉ࣭ԽΑΓ΋γΞ଎౓ͷۉ࣭Խͷ΄͏ ͕ KH ෆ҆ఆੑʹ༩͑ΔӨڹ͕ڧͦ͏ ▶ ೪ੑͷ΄͏͕ΑΓޮՌతʹϊΠζͷ੒௕Λ཈͑ΒΕΔͱ༧૝Ͱ͖Δ τkh = λ(ρh + ρl) √ ρhρl|vx,h − vx,l| (16) ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13