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回転座標系での古典場の⽅程式 について。微分形式からの導出 2022年4⽉29⽇ 量⼦と古典の物理と幾何@オンライン DeepFlow株式会社 深川宏樹 [email protected] 1

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流体機械(シロッコファン) 2 https://www.youtube.com/watch?v=o1xFOIbhbzk Rotating-peg, r=10.5[m], 11.5[rad/s],Re=10^3. 圧力 速度

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シミュレーション結果 3 Rotating-peg, r=10.5[m], 11.5[rad/s],Re=10^3. 圧力 速度 https://www.youtube.com/watch?v=o1xFOIbhbzk

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回転系の流体の⽅程式 4 !" !# + " ⋅ ∇ " = − 1 * ∇+ + ,Δ" + (慣性力項) ! 時間項 移流項 圧⼒項 粘性項

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コリオリ⼒と遠⼼⼒ 5 慣性⼒= −21×" − 1×(1×3) ! −2#×% −#×(#×') % ' コリオリ⼒ 遠⼼⼒ 回転座標系で中⼼に向かって移動

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コリオリ⼒と遠⼼⼒ 6 ! −#×(#×') % ( −2#×% = 2#×(#×') 慣性系で静⽌＝回転座標系で回転⽅向とは逆向きに移動 %=−#×' コリオリ⼒ 遠⼼⼒

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回転系の流体の⽅程式(通説) 7 !" !# + " ⋅ ∇ " = − 1 * ∇+ + ,Δ" − 21×" − 1×(1×3) コリオリ⼒ 遠⼼⼒ ! −2#×% −#×(#×') % '

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回転座標系のナビエ・ストークス 回転座標系のナビエ・ストークス⽅程式の導出を⾒たことがない… 8 11 (2021) 100096 Derivation of Navier–Stokes equation in rotational frame for engineering flow analysis Sananth H. Menon *, Ramachandra Rao A, Jojo Mathew, Jayaprakash J Solid Propulsion & Research Entity, Vikram Sarabhai Space Centre, ISRO, India A R T I C L E I N F O Keywords: Navier–Stokes equation Rotational frame Coriolis force A B S T R A C T One of the most frequently used governing equations underpinning engineering flow analysis is the renowned Navier–Stokes (NS) equation. Several references regarding the derivation of equation of motion in Cartesian coordinates are available in standard textbooks. However, derivation of above equation in rotational frame is missing in literatures. Flow analysis using NS equation in rotational frame is a prerequisite for analysis of various engineering problems like rotational flow dynamics in chemical reactors, lubricating oil behavior in various rotating machines, electrolyte flow behavior in electrochemical reactors with rotating electrodes, etc. Systematic understanding of each terms in the equation is essential to develop a suitable governing mathematical model of any physical flow problem with various level of complexities. This is possible only if same can be derived from fundamentals, detailing terms behind the equation. A systematic approach is made here to derive NS equation in rotational frame from basic Cartesian form. Contents lists available at ScienceDirect International Journal of Thermofluids journal homepage: www.sciencedirect.com/journal/international-journal-of-thermofluids One of the most frequently used governing equations underpinning engineering flow analysis is the renowned Navier–Stokes (NS) equation. Several references regarding the derivation of equation of motion in Cartesian coordinates are available in standard textbooks. However, derivation of above equation in rotational frame is missing in literatures. 計算がたいへん。 幾何的な意味を追うのが難しい。

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回転系での移流項 9 ! % ( "! = " + 1×3 !"′ !# + "′ ⋅ ∇ "! = !" !# + " ⋅ ∇ " +21×" + 1× 1×3 慣性系 回転系 ↑ほんと？ 微分形式で導出してみた。

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微分形式でのナビエ・ストークス⽅程式 10 !"! !# + ℒ"! − 1 2 d7"! "! = − 1 * d+ + ,Δ"’ !"′ !# + "′ ⋅ ∇ "′ = − 1 * ∇+ + ,Δ"′

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移流項 11 ℒ"! − 1 2 d7"! "! = 7"!d + 1 2 d7"! "! = 7"!d"! + 1 2 dg "!, "! = −"!×rot "! + 1 2 grad ?!# ℒ"! = 7"!d + d7"! リー微分

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微分形式 外微分 d 12 grad f: d# = %# %&! d&! + %# %&" d&" + %# % d rot +: d - $ # (/$ 0&$) = - %,$ # %/$ %&% d&% ∧ 0&$ = %/' %& − %/( %4 d& ∧ 04 + ⋯ d" d# d" ∧ d#=−(d# ∧ dz) ⼀形式 + = ∑ % /% d&%

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微分形式 内部積 !7 8 とリー微分 ℒ7 8 13 ベクトル場 7 8 = 9% % %&% ⼀形式 + = ∑ % /% d&% 内部積 :) * + = ∑ %+! # 9% /% :) * + ∧ ; = :) * + ∧ ; − :) * ; ∧ + リー微分 ℒ, - + = :) * d+ + d:) * + = −8×rot + + grad 8, +

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微分形式 ホッジ作⽤素 14 ∗: A形式→(3−k)形式 ∗ 1 = d& ∧ d4 ∧ dC ∗ dC = d& ∧ d4 ∗ (d& ∧ d4) = 0C ∗ (d& ∧ d4 ∧ dC) = 1 ∗∗= 1 d" d# d" ∧ d# d& =∗ (d" ∧ d#)

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移流項 15 ℒ "$% − 1 2 d7 "$% " + @ = ℒ& "" − 1 2 dg ", " +7& "d@ − 1 2 dg @, @ +ℒ& %" + ℒ& %@ 慣性⼒項 "! = " + A 慣性系 回転系 D = E0C % ( + , +

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移流項 16 + = "×$ =∗ , ∧ - =∗ .d& ∧ /d/ = ./!d0 "! = " + A 慣性系 回転系 D = E dC % ( + , + d/ /d0 d/ ∧ /d0 d& =∗ (d/ ∧ /d0)

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慣性⼒ 17 コリオリ⼒ 7& "d@ = 7& "d BC# dD = 27& " B dC ∧ CdD = 2(B?'CdD − B?(dC) = 27& "(∗ 1) (= 21×") Lwv ˜ = (dÿw + ÿwd) v = d(r 2 Êv ◊) + ÿwdv = d(r 2 Êv ◊) + ÿw 1 2rv ◊ dr · d◊ + r 2 dv ◊ · d◊ + dv r · = r 2 Êdv ◊ + 2rÊv ◊ dr ≠ 2rÊv ◊ dr ≠ r 2 Êdv ◊ + Ê A r 2 = Ê A ˆ(r 2 v ◊) ˆ◊ d◊ + ˆv r ˆ◊ dr + ˆv z ˆ◊ dz B Lvw ˜ = Lv(Êr 2 d◊) 2 遠⼼⼒ 1 2 dg @, @ = 1 2 d B#C# = B#C dC =∗ (1 ∧ @)(= 1× 1×3 ) D = E dz % ( + , + " = ?'dC + ?(CdD + ?)dz @ = BC#dD =∗ 1 ∧ CdC

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慣性⼒ 18 Lwv ˜ = (dÿw + ÿwd) v = d(r 2 Êv ◊) + ÿwdv = d(r 2 Êv ◊) + ÿw 1 2rv ◊ dr · d◊ + r 2 dv ◊ · d◊ + dv r · = r 2 Êdv ◊ + 2rÊv ◊ dr ≠ 2rÊv ◊ dr ≠ r 2 Êdv ◊ + Ê A r 2 = Ê A ˆ(r 2 v ◊) ˆ◊ d◊ + ˆv r ˆ◊ dr + ˆv z ˆ◊ dz B Lvw ˜ = Lv(Êr 2 d◊) 2 ℒ& %" = 7& %d" + d7& %" = −@×rot " + grad @, " = B !?' !D dC + !?( !D CdD + !?) !D Hz Lvw ˜ = Lv(Êr 2 d◊) = (dÿv + ÿvd) (Êr 2 d◊) = d(r 2 Êv ◊) + ÿvd(Êr 2 d◊) = d(r 2 Êv ◊) + ÿv2rÊdr · d◊ = r 2 Êdv ◊ + 2rÊv ◊ dr ≠ 2rÊv ◊ dr + 2rÊv r d◊ = r 2 Êdv ◊ + 2rÊv r d◊ 1 2dg(v + w, v + w) = 1 2d(g(v, v) + 2r 2 v ◊ Ê + r 2 Ê 2 ) = 1 2dg(v, v) + r 2 Êdv ◊ + 2v ◊ Êrdr + Ê 2 rdr ɺ (ˆt + Lu) u ˜ ≠ 1 dg(u, u) " = ?'dC + ?(CdD + ?)dz @ = BC#dD =∗ 1 ∧ CdC ℒ %&' − 1 2 d& %&' ' + ) = ℒ( % ' − 1 2 dg ', ' +&( % d) − 1 2 dg ), ) +ℒ( ' ' + ℒ( ' ) 慣性⼒項

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回転座標系のナビエ・ストークス 19 !" !# + ℒ& " − 1 2 d7& " " = − 1 * d+ + ,Δ"’ −7& " d@ + 1 2 dg @, @ − ℒ& %" コリオリ⼒ 遠⼼⼒ !

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まとめ 微分形式を使うと任意の座標系の計算が簡単に直接できる。 慣性⼒は、もう⼀つ項があった。 20 !" !# + ℒ& " − 1 2 d7& " " = − 1 * d+ + ,Δ"’ −7& " d@ + 1 2 dg @, @ − ℒ& %" コリオリ⼒ 遠⼼⼒ "×rot ' − grad ", ' −2.×' − .×(.×0) 2 = .×0 ) *+! *, d. + *+" *, .d, + *+# *, 0z