Some Studies on Spatial Dynamics of Incompressible Two Dimensional Wakes of Cylinders with Deformable Boundaries XiLin XIE. Yu Chen Department of Mechanics Engineering Science, Fudan University, Shanghai, CHINA Coordinates and Governing Equation Local/ Boundary Dynamics x( t)sr(*/)+s(* /)+r(mx, j-s(x, n)-n(,r) 人入A domain of b vorticity stream-function with respect to the curvilinear coordinates including time explicitly (x)=(9 distribution b Pressure Equation for Incompressible flows with respect to the curvilinear coordinates including time explicitly Verification of Algorithm As an verification of the whole algorithm, we reproduced the suppression of vortex stress of a circular der by introduce certain traveling wave on the cylinder's boundary as reported firstly by CJ Wu et al [21 Furthermore, we discovered that the same phenomenon can also occur on an elliptic cylinder as shown in of max I Figure: Snaps of flow patterns of an elliptic cylinder with traveling wave boundary. The ecs espectively in the case of Re=500. Global Dynamics a(t)=an sin(2rft), b(t)=bo sin(2rft-3) r(t)=ro+a sin(2ft)cos(6e) Contrastive Studies on Spatial Dynamics Distribution Global Dynamics Cylinder Type Flow Patterns Mean value Amplitude Spectrum Spe ROCC Distribution ,,, POEC Tearing, shedding.31 652P/Pf±qfo wOCCTearing, shedding Pf1±qf Local/ Boundary Dynamics Shear Stress Cylinder Type Vorticity Flux Major Angle of Strain Spectrum Analysis Time domain local varying ROCC pfi±qfo POEC P WOCC Pf1±qo deformation Distribution principle-axis-oscillation elliptic cylinder; wocC to stationary-wave-oscillation circular cylinder. fo &f. denote shedding frequency and boundary oscillation frequency respectively, p denotes nature number. onclusion A novel kind of vorticity stream-function algorithm for planar incompressible flows with deformable boundaries is put forward based the finite deformation theory with respect to the curvilinear coordinates stribution orresponding to the current physical configurations including time explicitly. As some applications, the spatial dynamics with respect to different kinds of deformable boundaries have been analyzed in the point of view of vorticity and vortex dynal Reference Snaps of flow patterns/distributions of stream-function corresponding to the principle-axis-oscillation elliptic der(left)and stationary-wave-oscillation circular cylinder(right)respe 1. Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56: 432-456 2. C.J. Wu, L. Wang and J.Z. Wu. Pression of Von Karman vortex street behind a circular cylinder by http://jpkc.fudaneducn/s/353/main.htm xiexilinofudan. edu. cn
Some Studies on Spatial Dynamics of Incompressible Two Dimensional Wakes of Cylinders with Deformable Boundaries XiLin XIE, Yu CHEN Department of Mechanics & Engineering Science, Fudan University, Shanghai, CHINA Coordinates and Governing Equation I Vorticity & stream-function with respect to the curvilinear coordinates including time explicitly 4ψ , g ij ∂ 2ψ ∂xi∂xj (x, t) − Γ k ij ∂ψ ∂xk (x, t) = −ω ω˙ = ∂ω ∂t (x, t) + V i ∂ω ∂xi (x, t) − ∂X ∂t i (x, t) ∂ω ∂xi (x, t) = 1 Re g ij ∂ 2ω ∂xi∂xj (x, t) − Γ k ij ∂ω ∂xk (x, t) (x, t), where ∂X ∂t i (x, t) = ∂X ∂t , g i I Pressure Equation for Incompressible flows with respect to the curvilinear coordinates including time explicitly ∆p = − V − ∂X ∂t (x, t) ⊗ ∇ : (∇ ⊗ V ) Verification of Algorithm As an verification of the whole algorithm, we reproduced the suppression of Vortex stress of a circular cylinder by introduce certain traveling wave on the cylinder’s boundary as reported firstly by C.J Wu et al [2]. Furthermore, we discovered that the same phenomenon can also occur on an elliptic cylinder as shown in Fig.1. Figure: Snaps of flow patterns of an elliptic cylinder with traveling wave boundary. The left and right subplots corresponding to the spatial distributions of stream-function and vorticity respectively in the case of Re=500. Global Dynamics a(t) = a0 sin(2πf t), b(t) = b0 sin(2πf t − π 2 ) r(t) = r0 + a sin(2πf t) cos(6θ) Distribution of streamline Distribution of vorticity Distribution of pressure Distribution of pressure Distribution of pressure Snaps of flow patterns/distributions of stream-function corresponding to the principle-axis-oscillation elliptic cylinder (left) and stationary-wave-oscillation circular cylinder (right) respectively in the case of Re = 100. Local / Boundary Dynamics Temporal distribution of σwall Time domain of τ at θ = 30 Temporal distribution of ∂ω ∂n Time domain of flux at θ = 30 Temporal distribution of max stress angle Contrastive Studies on Spatial Dynamics Global Dynamics Cylinder Type Flow Patterns Cd CL Mean value Amplitude Spectrum Analysis Spectrum Analysis SCC shedding 1.35 0.14 f0 f0 ROCC shedding 2.05 9.50 pf0 pf1 ± qf0 POEC Tearing, shedding 2.31 6.52 pf0 pf1 ± qf0 WOCC Tearing, shedding 1.73 3.06 pf0 pf1 ± qf0 Local / Boundary Dynamics Cylinder Type Shear Stress Vorticity Vorticity Flux Major Angle of Strain Time domain analysis Spectrum Analysis Time domain analysis Time domain analysis SCC local varying f0 local varying 45 ◦ ROCC global qusi-periodic varying pf1 ± qf0 Semi-cylinder qusi-periodic 45 ◦ POEC global qusi-periodic varying pf1 ± qf0 global qusi-periodic varying varying with deformation WOCC global qusi-periodic varying pf1 ± qf0 global qusi-periodic varying varying with deformation Remark: SCC referred to stationary circular cylinder; ROCC to radical-oscillation circular cylinder; POEC to principle-axis-oscillation elliptic cylinder; WOCC to stationary-wave-oscillation circular cylinder. f0 & f1 denote shedding frequency and boundary oscillation frequency respectively, p denotes nature number. Conclusion A novel kind of vorticity & stream-function algorithm for planar incompressible flows with deformable boundaries is put forward based on the finite deformation theory with respect to the curvilinear coordinates corresponding to the current physical configurations including time explicitly. As some applications, the spatial dynamics with respect to different kinds of deformable boundaries have been analyzed in the point of view of vorticity and vortex dynamics. Reference 1. Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56:432-456. 2. C.J. Wu, L. Wang and J.Z. Wu. Spression of Von Karman vortex street behind a circular cylinder by traveling wave generated by a flexible surface. Journal of Fluid Mechanics 574:365-391,2007. Created with LATEXbeamerposter http://jpkc.fudan.edu.cn/s/353/main.htm xiexilin@fudan.edu.cn