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Computation of the Starting Vortex Past a Flat Plate

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Computation of the Starting Vortex Past a Flat Plate XXIV ICTAM, 21-26 August 2016, Montreal, Canada COMPUTATION OF THE STARTING VORTEX PAST A FLAT PLATE Monika Nitsche1, Ling Xu2, and Robert Krasny∗2 1Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA 2Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA Summary A comparison is presented of numerical methods applied to compute the starting vortex past a flat plate. The methods include (1) direct numerical simulation (DNS) by a finite-difference scheme, and (2) a vortex sheet model. Preliminary results will also be presented for (3) DNS by a viscous vortex method. The advantages and disadvantages of the methods will be discussed. INTRODUCTION When a viscous flow passes a solid body, the flow sticks to the body and a boundary layer forms along the body surface. The boundary layer thickens due to viscous diffusion and it is convected downstream until eventually it separates and rolls up into a starting vortex. The starting vortex affects the lift and drag forces on the body, and this has implications in many fields such as aerodynamics and biolocomotion. Several groups have compared different models and numerical methods for computing this type of flow separation. For example a recent study compared a direct numerical simulation (DNS) with a discrete vortex model (DVM) for flow past an airfoil [9]. The results showed that the DVM captures some large scale features of the flow and was much less costly than the DNS, but the DVM was lacking in its treatment of the viscous boundary layer. Here we consider the starting vortex past a flat plate and compare results obtained using several numerical methods, (1) DNS by a finite-difference scheme, and (2) a vortex sheet model. Preliminary results will also be presented for (3) DNS by a viscous vortex method. The advantages and disadvantages of the methods will be discussed. PROBLEM DESCRIPTION A flate plate is immersed in a viscous fluid. The plate is held fixed on the interval −0:5 ≤ x ≤ 0:5 in the xy-plane. The incoming flow has speed U = tp and angle of incidence α. Here we consider the case of impulsively started flow p = 0 and normal incidence α = 90◦, and we will report on other cases at the ICTAM. We consider the Navier-Stokes equations in vorticity form, −1 2 @t! + u · r! = Re r !; (1) where ! is the vorticity, u is the velocity, and Re is the Reynolds number. The velocity is obtained from the stream function, 2 u = ( y; − x), where the vorticity and stream function are related by the Poisson equation r = −! with boundary conditions = 0 on the plate surface, and ! 1 for jxj ! 1. The no-slip condition u = 0 on holds on the plate surface. NUMERICAL METHODS DNS by a finite-difference scheme The Navier-Stokes equations are solved by operator splitting, where the inviscid and viscous parts are computed in two successive stages per time step. The inviscid part, @t! + u · r! = 0, is computed by a semi-Lagrangian method, and the −1 2 viscous part, @t! = Re r !, is computed by the Crank-Nicolson method. Details are in [8, 12, 13]. Vortex sheet model The vorticity distribution has two components, a free vortex sheet that is shed at the edge of the plate, and a bound vortex sheet on the plate. The free sheet is represented by regularized point vortices (vortex blobs) with smoothing parameter δ, and the bound sheet is represented by singular point vortices. The vortex blobs are advected using the Biot-Savart integral. The bound vortex sheet strength is computed by solving an integral equation on the plate. The unsteady Kutta condition is imposed at the edge of the plate to account for circulation shedding. Details are in [7]. DNS by a viscous vortex method Lighthill [5] advocated that the vorticity should be used in DNS of viscous flow past a solid body, where the no-slip condition can be satisfied by creating vorticity on the body surface, and then allowing it to diffuse and convect. This approach has been implemented in various ways by many investigators (see for example [2, 11, 3]). In the present work we implement this approach using recently developed adaptive refinement and remeshing techniques for accuracy [4, 1] and a treecode algorithm for efficiency [6]. This is current work and we will report on our preliminary results at the ICTAM. ∗Corresponding author. Email: [email protected] NUMERICAL RESULTS Figure 1 compares results obtained using (1) DNS by a finite-difference scheme with several values of Reynolds number Re, and (2) a vortex sheet model with several values of the smoothing parameter δ. The DNS results show a streakline of passive particles introduced at the edge of the plate, and the vortex sheet results also show a streakline, but in this case these are the computed vortex blob locations. The streaklines roll up into spirals that become tighter as Re ! 1 and δ ! 0. There Outline is good agreement between the DNS and vortex sheet model in terms of the size, shape, and location of the spiral. Similar results were previously obtained for the case in which a free shear layer rolls up in unbounded flow [10], and the present work Comparisonis an extension with to Vortex the case sheet in which – the Streaklines, flow separatesp at= the 0 edge of a solid body. Figure 1 also presents the shed circulation Γ as a function of time, again showing good agreement between the two models. Re=250 Re=500 Re=1000 Re=2000 0.9 y −0.15 0 0.90 0.90 0.9 0x 0.9 x x x δ=0.2 δ=0.1 δ=0.05 δ=0.025 0.9 y −0.15 0 0.90 0.90 0.9 0x 0.9 x x x 1 / 5 Figure 1: Starting vortex past a flat plate, comparison of numerical results, (left) streaklines of separating shear layer, (top left) DNS by a finite-difference scheme with Reynolds number Re, (bottom left) vortex sheet model with smoothing parameter δ, (right) shed circulation Γ as a function of time. CONCLUSIONS We computed the starting vortex in viscous flow past a flat plate using several numerical methods, direct numerical sim- ulation (DNS) and a vortex sheet model. There is good agreement between the DNS and vortex sheet results in terms of the large scale features of the flow, namely the size, shape, and location of the spiral core, and the shed circulation as a function of time. The vortex sheet model is much less costly than the DNS, but it has limited applicability due to the omission of the boundary layers present in the true viscous flow. We are currently developing a viscous vortex method which should address the limitations of the vortex sheet model. References [1] Bosler P., Wang L., Jablonowski C., Krasny R.: A Lagrangian Particle/Panel Method for the Barotropic Vorticity Equations on a Rotating Sphere. Fluid Dyn Res 46:031406, 2014. [2] Chorin A.J.: Numerical Study of Slightly Viscous Flow. J. Fluid Mech 57:785–796, 1973. [3] Cottet G.-H., Koumoutsakos P., Vortex Methods: Theory and Practice. Cambridge University Press, 2000. [4] Feng H., Kaganovskiy L., Krasny R.: Azimuthal Instability of a Vortex Ring Computed by a Vortex Sheet Panel Method. Fluid Dyn Res 41:051405, 2009. [5] Lighthill M.J.: Introduction, Boundary Layer Theory, in: L. Rosenhead (Ed.), Laminar Boundary Layers. Oxford University Press, pp. 54–61, 1963. [6] Lindsay K., Krasny R.: A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow. J. Comput Phys 172:879–907, 2001. [7] Nitsche M., Krasny, R.: A Numerical Study of Vortex Ring Formation at the Edge of a Circular Tube. J. Fluid Mech 276:139–161, 1994. [8] Nitsche M., Xu L.: Circulation Shedding in Viscous Starting Flow Past a Flat Plate. Fluid Dyn Res 46:061420, 2014. [9] Ramesh K., Gopalarathnam A., Granlund K., Ol M.V., Edwards J.R.: Discrete-Vortex Method with Novel Shedding Criterion for Unsteady Aerofoil Flows with Intermittent Leading-Edge Vortex Shedding. J. Fluid Mech 751:500–538, 2014. [10] Tryggvason G., Dahm W.J.A., Sbieh K.: Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation, J. Fluids Engin 113:31–36, 1991. [11] Wu J.Z., Wu J.M.: Boundary Vorticity Dynamics Since Lighthill’s 1963 Article: Review and Development. Theor Comput Fluid Dyn 10:459–474, 1998. [12] Xu L., Nitsche M.: Scaling Behaviour in Impulsively Started Viscous Flow Past a Finite Flat Plate. J. Fluid Mech 756:689–715, 2014. [13] Xu L., Nitsche M.: Start-Up Vortex Flow Past an Accelerated Flat Plate. Phys Fluids 27:033602, 2015..
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