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A Turbulent Flow over a Square Cylinder with Prescribed and Autonomous Motions

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The present study focuses on the development of the methodology necessary for the study of fluid-structure interactions between a square cross-section cylinder and a turbulent cross-flow. The square cylinder is simulated in three configurations: 1)
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  Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 4, pp. 573–586 (2009) A TURBULENT FLOW OVER A SQUARE CYLINDER WITHPRESCRIBED AND AUTONOMOUS MOTIONS J. Hines* + , G.P. Thompson** and F.S. Lien**  Department of Mechanical Engineering, University of Waterloo,200 University Ave. W., Waterloo, ON, N2L 3G1, Canada +    E-Mail: jhines@uwaterloo.ca ( Corresponding Author  )   **  Motioneering Inc., 650 Woodlawn Road West, Guelph, ON, N1K 1B8, Canada ABSTRACT: The present study focuses on the development of the methodology necessary for the study of fluid-structure interactions between a square cross-section cylinder and a turbulent cross-flow. The square cylinder issimulated in three configurations: 1) stationary, 2) undergoing prescribed vertical motion, and 3) undergoingautonomous vertical motion. Motion of the square cylinder through the computational domain is facilitated through amethod of localized mesh deformation, which deforms the control volumes immediately above and below the squarecylinder as it moves. Results for prescribed vertical motion of the cylinder, governed by sinusoidal equations of motion, are compared to those from published literature and a substantial agreement is achieved. In order to simulateautonomous motion, a vibration model is implemented providing direct application of the shear and pressure balancearound the square cylinder to the motion of the cylinder. Keywords: bluff body, autonomous motion, prescribed motion, mesh deformation, unsteady RANS 1.   INTRODUCTION Any object immersed in a flow is subjected to pressure and shear loading which will create aforce imbalance on the object. This forceimbalance and resulting deflection or motiondeserves study as it is a commonly encountered phenomenon. Considerable research has been performed on both square and circular cylinderssubjected to a cross-flow. Okajima (1982) studiedsquare cylinder cross-sections in a cross-flow andcharted the Strouhal numbers of these geometriesunder multiple flow conditions. Franke, Rodi andSchönung (1990), Davis and Moore (1982), andSohankar, Norberg and Davidson (1997) performed experimental or numerical research onstationary square cylinders at low Reynoldsnumbers. Lyn and Rodi (1994), and Lyn et al.(1995) performed extensive experimental work ona stationary square cylinder at a Reynolds number of 21,400. However, few studies have been performed on square cylinders undergoingsinusoidal motion in a vertical plane whilesubjected to a cross-flow. Further, even fewer studies have permitted the cylinder to move in anautonomous manner. Similar studies have been performed on circular cylinders with relevance toheat exchangers and electrical cables. However,square cylinders deserve similar study asrelevance can be extended to office towers or  bridge decks.The introduction of object motion to the unsteadyflow simulation requires consideration of themotion on the flow field and the methodologies of modelling this interaction. A method of deforming or otherwise translating the mesh isrequired to allow for an accurate allowance of model motion and communication of this motionto the flow solver and associated computationalroutines. Although prescribed motion can beaccommodated through an equation of motion,the introduction of autonomy requires feedback of the pressure and shear loading and a method of determining the corresponding motion.The present work presents a square cylinder undergoing either prescribed or autonomousvertical motion. The method of grid deformationand autonomy are presented to complete thework. 2.   TURBULENCE MODEL The instantaneous value of anyflow variable,  ,can be fundamentally decomposed into severaldistinct components. Here, the flow variable isdecomposed into '       (1)where     is the phase-averaged (or periodic)component and    is the random fluctuatingcomponent. The basic phase-averaged continuity  Received: 10 Jan. 2009; Revised: 21 May 2009; Accepted: 17 Jun. 2009 573  Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 4 (2009) and momentum equations for an unsteady,incompressible fluid flow written in Cartesiantensor notation are    Continuity: 0  j j u x  (2)    Momentum:   '' iijij j uuuuutx      1  jiijji u pu xxxx                (3)where '' 2,3  jiijtij ji uuuuk  xx               '' 12 ii kuu  (4)The most common of the two-equation Reynolds-averaged Navier Stokes (RANS) turbulencemodels, which relate the turbulent length andvelocity scales to eddy viscosity,   t  , is the highReynolds number k-  model. The transport of turbulent kinetic energy is modeled by   () t  jk  jj k ukP txxx                     kj k   (5)and the dissipation rate,  , is obtained through    j j utx           12 () t k  jj CPC kx                     x    (6)Here, v t  is the eddy viscosity defined as 2 t  k C        . The turbulence model constants areC  =0.09, C  1 =1.44, C  2 =1.92,  k  =1.0 and   =1.3.Finally, the production of turbulence kineticenergy,  P  k  ,   following the suggestion of Kato andLaunder (1993), can be written as k   PCS        ,where 2 1,2  ji ji uuk S     xx           2 12  ji ji uuk  xx          (7)The Kato-Launder model, used in the presentstudy, is implemented to reduce the excessive production of turbulent kinetic energy in thestagnation region found in front of the squarecylinder. As  is based on flow rotation, insimple shear flows, S  and  are equal. However in stagnation regions,   0, resulting in thedesired reduction of the turbulent kinetic energy production. The choice of the Kato-Launder model is discussed further in Section 6.2. 3.   NUMERICAL METHOD The present study was conducted with an adaptedversion of the STREAM code by Lien andLeschziner (1994). The STREAM code employs acurvilinear non-orthogonal co-located finitevolume algorithm. The SIMPLE algorithm byPatankar and Spalding (1972) is used to couplethe pressure and velocity fields. Rhie and Chow(1983) interpolation is utilized to avoid non- physical oscillations in the pressure field. TheQUICK convection scheme by Leonard (1979)was used for the numerical fluxes. The unsteadyReynolds averaged continuity, momentum andturbulence equations are solved iteratively byapplication of SIP by Stone (1968). Unsteadyterms are approximated by a fully implicitsecond-order accurate three-time-level method.The computational domain used throughout the present work is illustrated in Fig. 1. Although thegeometry being modelled is symmetric around ahorizontal centre axis, a full domain is employedto properly capture the vortex shedding phenomena. A half-size domain employingsymmetry cannot be used as the vortex shedding phenomena will not be captured. The domaindimensions chosen here are given in Table 1.All solid boundaries including the cylinder surfaces were provided with the wall-function boundary conditions. The inlet boundary was provided with a Dirichlet condition, and the outletemployed the convective boundary condition: 0. ii uuutx        574  Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 4 (2009) X YDD Y YOUTFLOWBOUNDARYUPPER (NORTH) WALLLOWER (SOUTH) WALLINFLOWBOUNDARYNORTHFACESOUTHFACEEASTFACEWESTFACE tl X d X u  Fig. 1 Computational domain.Table 1 Domain dimensions.D X u X d Y l Y t  1 4.5 15 6.5 14 -2.000-1.500-1.000-0.5000.0000.5001.0001.5002.000050100150200250300 Time    S  e  c   t   i  o  n  a   l   L   i   f   t   C  o  e   f   f   i  c   i  e  n   t   (   C    l    )   Fig. 2 Sectional lift coefficientdimensionless time historyfor a stationary squarecylinder.Table 2 Comparison of the present results with previously published results. St C d   Present Course Mesh 0.141 2.15Present Fine Mesh 0.143 2.29Lyn et al. (1995) 0.132 2.10Bosch and Rodi (1995) 0.146 2.11Kato and Launder (1993) 0.145 2.05 4.   STATIONARY CYLINDER RESULTS Prior to introduction of the motion algorithms, theSTREAM code was benchmarked for flow over astationary square cylinder at a Reynolds number of 21,400. This case was selected because of thewide selection of previously published resultsincluding the experimental results of Lyn et al.   (1995), and the numerical, URANS results of Bosch and Rodi (1995) and Kato and Launder (1993).The benchmark results from the present study areshown in Table 2. These results, obtained with acoarse mesh of 100×70 nodes and a fine mesh of 200×140 nodes, compare well with the numericalresults of Bosch and Rodi and Kato and Launder,when the Kato Launder modification to the k- ε  model is employed. The study also compares wellto the experimental results of Lyn et al.   Thedevelopment of an oscillatory flow is illustrated 575  Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 4 (2009)  by the time history of the sectional lift coefficientshown in Fig. 2.Further parametric studies including sensitivity of St (Strouhal number) and C  d  (drag coefficient) to    =  D / Y  t    (blockage ratio),  X  u (the distance fromthe inlet to the cylinder) and Δ t *= Δ tu  / D  (dimensionless time step) are conducted in order to isolate the numerical uncertainties from errorsassociated with RANS turbulence modeling, andresults are given in Tables 3–5. As shown inTables 3–4, the sensitivities of the predictedand C  d  to    , in the range of 5% to 8%, and to  X  u , in the range of 4.5 to 8.5, are low. Therefore,    =7.1% and  X  u =4.5 were used in the presentstudy. Similarly, Table 5 indicates that the predicted and C  d  ,   were reasonably insensitiveto three different dimensionless time steps (i.e., Δ t *=0.05, 0.025, 0.0125) used. Thus, for theremainder of the simulations, Δ t *=0.025 wasused. St St    Table 3 Effect of   on St and C d for a flow over astationary cylinder at Re=21,400. St C d    =5.0%0.143 2.28  =7.1%0.144 2.29  =8.0%0.144 2.29 Table 4 Effect of X u on St and C d for a flow over astationary cylinder at Re=21,400. St C d  X u =4.5 0.144 2.29X u =6.5 0.146 2.25X u =8.5 0.144 2.21 Table 5 Effect of   t* on St and C d for a flow over astationary cylinder at Re=21,400. St C d    t*=0.050.144 2.29  t*=0.0250.144 2.29  t*=0.01250.144 2.29 5.   MESH DEFORMATION TECHNIQUE The motion of the square cylinder requires atechnique to communicate the cylinder positionand velocity with the flow solver. A method of locally deforming the mesh, referred to as the“local mesh deformation” approach, wasincorporated. Figure 3b shows the rationale of the Fig. 3 Mesh deformation methodology. 576  Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 4 (2009) 577 methodology with only a single mesh layer of deformation, where the top surface of the j indexcell is deformed upwards to the top of the squarecylinder, forming a layer of thinner cells directlyabove the square cylinder, and a layer of thicker cells within the bounds of the cylinder. The twocells adjacent to the corners of the cylinder aredeformed into trapezoids. Although this is themost simplistic method for local meshdeformation, serious problems can occur whenthe top surface of the square cylinder is near, butdoes not coincide with, a mesh line. As the topsurface of the square cylinder nears the topsurface of a j+1 index cell, and the top surface of the adjacent j index cell is deformed to match, alayer of very high-aspect-ratio cells are formed, asdepicted in Fig. 3c. Here the aspect ratio isdefined as the ratio of the length to the width of the cell. Elongated cells with high-aspect-ratioscan cause a loss of resolution along the length(streamwise direction) of the cell, ultimatelyreducing the accuracy of the solution.In order to overcome the difficulties associatedwith very high-aspect-ratio cells, a modificationto this method is proposed, which propagates themesh deformation across multiple layers of cellsin the mesh. Figure 3d shows a method involvingthe deformation of two layers of cells above thecylinder. The deformation of the top surface of the j index cell is such that its new position is co-located with the top of the cylinder,  y top . Thedeformation of the top surface of the j+1 indexcell is such that it is midway between the top of the cylinder and the top surface of the j+2 indexcell as described by Eq. (8) below: 12 (,)(,1)(,2) toptop  yijy yijyijy       (8)Using this method, the aspect ratio of thedeformed mesh can change a maximum of 200%compared to the aspect ratio of the srcinal mesh.This deformation is well within the capabilities of the STREAM flow solver, and does not result in asignificant loss in the accuracy of the solution. Aconsequence of this method of mesh deformationis that cells will be uncovered as the cylinder moves through the mesh as shown in Fig. 4. In the present study it was found that the verticalvelocity of the uncovered cells could be set toeither the velocity of the cylinder or the velocityof the lower adjacent fluid cell as shown in Fig. 4with practically no change in the results. This is to be expected, as vertical velocity of the flow incells above and below the cylinder should always be nearly identical to the velocity of the cylinder.Modification of the boundary conditions is alsorequired for cylinder motion. For example, on theright (back) or left (front) face of the squarecylinder, the wall shear stress   w   can beevaluated by    1/21/4 (), for 11.6(), for 11.6ln() ww vyt  x xkCvyt  x Ex              (9) SQUARECYLINDER v=v(i,j)   v=y(t) CYLINDERSQUARE V=? Previous Time Step: t o Current Time Step: t V=?V=? v=v(i,j)v=v(i,j)v=v(i,j)v=v(i,j)v=v(i,j)v=y(t)v=y(t)v=y(t)v=y(t)v=y(t)   Fig. 4 Uncovered cells due to movement of square cylinder through the mesh.
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