U Fig. : Physical state Navier-Stokes u t +(u grad)u = ρ gradp + ν u, () divu =. () u p ρ ν ω =(,,ω) b ω t +(u grad) ω = ν ω. (3) u Biot-Savart u = ω(
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1 B- Transient Flows around an Impulsively Started Rectangular Cylinder by a Hybrid Vortex Method (Combination with Remeshing Method) noda@fluid.energy.osakafu-u.ac.jp ueda@fluid.energy.osakafu-u.ac.jp tamano@fluid.energy.osakafu-u.ac.jp kida@energy.osakafu-u.ac.jp Kensei NODA, Osaka Prefecture University, Gakuencho, Sakai-shi, Osaka 99 83, Japan Yoshiaki UEDA, Osaka Prefecture University, Gakuencho, Sakai-shi, Osaka 99 83, Japan Hirokazu TAMANO, Osaka Prefecture University, Gakuencho, Sakai-shi, Osaka 99 83, Japan Teruhiko KIDA, Osaka Prefecture University, Gakuencho, Sakai-shi, Osaka 99 83, Japan The flow around a rectangular cylinder is one of typical flows with fixed separation and it is important to know the aerodynamic forces from the engineering point of view. It was reported by the experimental works that cutting the edges of the rectangular cylinder resulted the reduction of the drag force, however, the detailed numerical simulation has not been reported for the effect of cutting edges of the rectangular cylinder. The present paper constructs two hybrid vortex methods, () the vortex sheet method and the vortex blob and () the vortex sheet method and the method of using the local smoothing interpolation function proposed by Cottet. The numerical results of the rectangular cylinder without cutting by the two methods, which is started impulsively with a constant speed, are compared and it is shown that to use the interpolation function affects the increase of the viscosity. The flows around the rectangular cylinder with and without cutting their edges are simulated by the method ().. () () () ()(8) (7).8 ()(6)(7) Davis & Moore () 8 Okajima et al. (9).. 6 Hwang & Sue (3) Barton () Simple () Sarpkaya & Ihrig (6) Chiu &Ko (8) Otsuka et al. (9) Vortex sheet Vortex blob Shirato & Matumoto () Vortex-in-cell Vortex blob & sheet Cottet. Copyright c by JSCFD
2 U Fig. : Physical state Navier-Stokes u t +(u grad)u = ρ gradp + ν u, () divu =. () u p ρ ν ω =(,,ω) b ω t +(u grad) ω = ν ω. (3) u Biot-Savart u = ω(x ) (x x ) π x x 3 dx + u p. () u p rot u p = () u p random-walk Kida et al. () σ γ γ Kelvin 3. Vortex blob & sheet Vortex blob Chorin () Cottet & Koumoutsakos (3) Chorin a Vortex sheet Chorin () cut-off 3.3 Remeshing Remeshing Cottet (3) x p ˆγ q, ˆx q Λ(x) γ p γ p (x p,τ)= q ˆγ q (ˆx q,τ)λ ( ) xp ˆx q, () h h ( ) (6a) Λ i (x) (6b) Λ j (x) ( (3) ) ( x )( x), if x<, Λ i (x) 6 ( x)( x)(3 x), if x<,, otherwise. (6a), if x >, Λ j (x) ( x ) ( x ), if x, x + 3 x 3, if x. (6b) 3. v = u e n = on S, u = u e t = on S. e n, e t h i (x, y) x y (u, v) u (x, y) u u/ y Copyright c by JSCFD
3 v v/ x ξ (x, y) hi u(x, y) u p + ξ(x, z)dz y u p + ξ j d j. (7) y j y u p y y j y j y d j d j d j = x x j. (8) l l ξ/ (7) u(x k,y k )=u p + y j y ξ j d j + ξ k. (9) u = ξ c ξ c = (u p + j ξ j d j ). () ξ c ξ max ξ c >ξ max ˆξ c = ξ c n ξc n ξc = [ ξc ξ max ] for ξ c >ξ max. () [ ] Gauss ξ c ξ max ξ max ˆξ c n ξc random-walk Vortex blob & sheet Γ Γ=ξ l. () Γ Remeshing (6a) (6b) Vortex blob & sheet Kida & Kurita () 3. Random walk random walk r Ω N(, ) r = t Re Ω. (3) Re x, y. ()(6) Navier Stokes H = (u ω), () H =(p p )/ρ + ( u U ) / p Green H = J H π C n log x x ds + ν ω π s log x x ds, () C C n J J = { π x (vω) } y (uω) dx dy. (6) D D J J = Γ j π x x j j (u j(y y j ) u j (x x j )) ( exp( x x j /δ ) ) Γ [ j π δ E ( x x j /δ ) j ] E ( x x j /δ ), (7) (u j,u j ) j Z e x E (x) = x x dx (7) J () x H (). () Vortex blob & sheet method Re = 3 n exp( (n n +)/) n t =. n > n =8 ξ max =. Cut-off σ = l i /π l i h i =/Re / 3 Copyright c by JSCFD
4 () Remeshing.. Vortex blob & sheet. C Dp 3 t Fig. : Time history of the drag coefficient of the rectangular cylinder without cutting: Re = 3. Fig. : Vortex blob distribution by Vortex blob & sheet method: Re = 3 and t =.8363 C p 6 8 panel number Fig. 6: Pressure distribution on the surface of the rectangular cylinder without cutting: Re = 3 and t = Fig. 3: Vortices distribution by Remeshing method: Re = 3 and t =.8363 Remeshing C D C Dp. Okajima et al. () Davis et al. () 3 Davis et al. () t =8 C D t = 3 6 Fig. : Comparison of the vortices distributions between the two methods: Re = 3 and t =.8363 Vortex blob & sheet Remeshing 3. Vortex blob & sheet Remeshing 6 Fig. 7: Panel number Copyright c by JSCFD
5 6 Fig. : Vortex blob distribution around the rectangular cylinder with cutting by Vortex blob & sheet method: Re = 3 t =.8363 c /a = c /a =. 6 Fig. 8: Vortex blob distributions in the case of aspect ratio /3, /6 and 3, 6: Re = 3 and t = ルは等間隔であるので ここではパネル番号を横軸にとっ ている パネル番号は後面上端から番号が付けられてお り 従って 角柱背面と前面のパネル番号は と 6 である 図 7 参照 図 8 は矩形断面角柱の縦横比 b/a = /3, /6 および 3, 6 における渦核分布を描いたものである 縦横比が より 小さくなると 矩形端から流出した剥離渦が大きく広が り それらが不安定となり合体し大きな渦塊になる そ れらがさらに合体し後流を形成していく様子が良く分か る 一方 縦横比が より大きくなると 矩形前端で剥 離した剥離泡は側面に再付着し Okajima et al.() の数 値結果と定性的に同じ挙動を示すことが分かる. Fig. method: Re =, Left side: c = c =, Right side: c /a = c /a =. ズ数が. 3 であり 定常流れを可視化したものであ る 本計算は定常流れに至るまでの長時間計算をしてい ないので 直接の比較検討は出来ないが 定性的には本 結果は妥当であると推察される 図 は切落とした角部の速度ベクトルを示したもの である この図から分かるように 角部では強い循環流 れはなく 殆どよどんだ状態になっているが 側壁端に 近くなると流速が大きくなっている 従って 切落とし 部での圧力は一定ではない 切落とし端から剥離した流 れは切落とし部のよどみ流れを囲い込むように流れ 側 壁端付近で再付着し 再び剥離し後流へ流れている こ のため 剥離渦の強さが小さくなり 背圧も小さくなり それによって抵抗が軽減していると思われる この定性 的な結果は 倉田らの実験結果と良い一致を示している 切落とし矩形断面 c c : Flow patterns given by hydrogen bubble a Uw 結 言 急発進する二次元角柱まわりの流れを渦法によって数値 解析した ここで用いた渦法は 渦核と渦シートを併用す る方法 Vortex blob & sheet 法 と流れ場に格子を形成し 内挿関数を用い渦度場を格子点に貼り付ける Remeshing 法である 物体表面からの渦度の生成はスプリット法で 粘性拡散は random walk 法によった 以上の結果をまと めると以下のことが分かった: () 内挿関数を用いると 渦塊の拡散が大きい このことは粘性係数が増加した場 合に対応していると思われる () 従って Vortex blob & sheet 法が高レイノルズ数遷移流れをシミュレートす るにはより適切と思われる 次に 矩形柱の前方角部を切落とした場合について Vortex blob & sheet 法によって数値計算した その結果 次 6. b Fig. 9: Rectangular cylinder with cutting 矩形柱の結果から Vortex blob & sheet 法の結果は妥 当であると推察されるので その方法によって切落とし のある矩形柱について数値計算した 図 9 は切落としの ある角柱の形状を示す 図 は正方形角柱の前方の角部 を切落とした場合の渦核分布である 図 と比較すると 良く分かるように 角部を切落とすとカルマン渦列への 移行は遅れる 従って 後流の長さは長くなり また 後 流の幅も小さくなる この計算時間内ではカルマン渦列 への移行が明確ではない 図 は倉田ら (7) による水 素気泡法による実験結果である この実験ではレイノル c by JSCFD Copyright
6 Fig. : Velocity field in the cutting region of the edge of the rectangular cylinder: Re = 3, t = , c /a = c /a =. () () (6) (8) C (678) (),,, B-7, (98), pp () Davis, R.W. & Moore, E.F., A numerical study of vortex shedding from rectangles, J. Fluid Mech., 6 (98), pp (3) Okajima, A., Strouhal number of rectangular cylinders, J. Fluid Mech., 3 (98), pp (),,, 33, (986), pp (),,,, 3, (986), pp. 99. (6) Sarpkaya, T. & Ihrig, C.J., Impulsively started steady flow about rectangular prisms: Experiments and discrete vortex analysis, Trans. ASME, J.Fluids Engr., 8 (986), pp. 7. (7),,,,, 9, (99), pp. 3. (8),,,, 38, A(99), pp (9) Okajima, A., Ueno, H. & Sakai, H., Numerical simulation of laminar and turbulent flows around rectangular cylinders, Int. J. Num. Methods in Fluids, (99), pp () Ozono, S., Ohya, Y. & Nakamura, Y., Stepwise increase in Strouhal number for flows around flat plates, Int. J. Num. Methods in Fluids, (99), pp. 36. () Ohya, Y., Nakamura, Y., Ozono, S., Tsuruta, H. & Nakayama, R., A numerical study of vortex shedding from flat plates with square leading and trailing edges, J. Fluid Mech., 36 (99), pp. 6. () Johansson, S.H., Davidson, L. & Olsson, E., Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a k-ε turbulence model, Int. J. Num. Methods in Fluids, 6 (993), pp (3) Hwang, R.R. & Sue, Y.C., Numerical simulation of shear effect on vortex shedding behind a square cylinder, Int. J. Num. Methods in Fluids, (997), pp. 9. () Barton, I.E., Comparison of simple- and piso-type algorithms for transient flows, Int. J. Num. Methods in Fluids, 6 (998), pp (),,, B-6, (998), pp (6) 3,,, 7, (999), pp (7) 3,,, 7, (999), pp. 3. (8) Chiu, A.Y.W. & Ko, N.W.M, Numerical simulation of proximity interference of two square cylinders, ASME/JSME Fluids Engr. Coference, (999). (9) Otsuka, M., Kida, T., Wada, M. & Kurata, M, Two-dimensional transient flows around rectangular cylinder by a vortex method, Vortex Methods, ed. Kamemoto, K. & Tsutahara, M., World Scientific, Singapore, (), pp. 6. () Shirato, H. & Matsumoto, M, Some application of vortex method to wind engineering problem, Vortex Methods, ed. Kamemoto, K. & Tsutahara, M., World Scientific, Singapore, (), pp.. () Kida, T., Nagata, T. & Nakajima, T., Accuracy of the panel method with distributed sources applied to two-dimensional bluff bodies, CFD Journal, (993), pp () Chorin, A.J., Numerical study of slightly viscous flow, J. Fluid Mech., 7 (973), pp (3) Cottet, G.-H. & Koumoutsakos, P.D., Vortex Methods, Cambridge, (). () Chorin, A.J., Vortex sheet approximation of boundary layer, J. Comp. Phys., 7 (978), pp. 8. () Kida, T.& Kurita, M., High Reynolds number flow past an impulsively started circular cylinder (Time 6 Copyright c by JSCFD
7 marching of random walk vortex method), CFD Journal, (996), pp (6),,,,, B-63 (997), pp (7) Kida, T., Ueda, H. & Kurata, M., Pressure distribution of transient flow around an impulsively started two-dimensional elliptic cylinder by a vortex method, CFD Journal, 9 (), pp Copyright c by JSCFD
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