Parallel vorticity formulation for calculating lift and drag on bluff bodies in a free stream

Transcription

1 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) Parallel vorticity formulation for calculating lift and drag on bluff bodies in a free stream D G, Tiptonl, M S Ingberl &M J Brown2 lengineering Sciences Center, Sandia National Laboratories, New Mexico, USA Department of mechanical Engineering, University of New Mexico, USA ABSTRACT: A parallel vorticity code is developed to study flows over bluff bodies in a free stream The vorticity method has certain advantages over primitive variable methods for incompressible flows including a reduced equation set through the elimination of the pressure variable However, because pressure is eliminated, the calculation of aerodynamic coefficients such as lift and drag must be performed in postprocessing In the current research, an efficient method for recovering the pressure field locally on a body in a free stream is developed, In order to analyze large wake regions, a parallel implementation of the vorticity code is developed The vorticity code is based on a three-step algorithm The first step involves determining vortex sheet strengths on the surface of the body which yields Neumann boundary conditions for the vorticity equation, The second step involves determining the interior velocity field using the generalized Helmholtz decomposition representing the flow kinematics The third step involves solving the vorticity equation, Each step requires a separate parallel strategy The parallel efficiency of the overall algorithm is evaluated by considering the benchmark problem of flow over a cylinder

2 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) 106 Applications of High-Performance Computing in Engineering VII 1 Introduction The vortex street generated by flows about a cylinder placed in a freestream is one of the more interesting problems of classical fluid mechanics The associated unsteady forces experienced by the body have been extensively studied since von K4rm& s [8] analysis in 1911 Several classical experimental studies have been performed to calculate the average pressure distribution and drag on submerged bodies in the range of Reynolds numbers between 100 and 1000 [13, 14, 16] Numerous numerical studies have also been performed to calculate lift and drag forces on a submerged cylinder in a free stream Some of the numerical analyses have been based on finite difference methods (FDM) [7, 10], finite element methods (FEM) [4, 2], and boundary element methods (BEM) [1, 3] Vorticity methods have certain advantages over primitive-variable methods for this class of problems including a reduced equation set through the elimination of the pressure variable, identical satisfaction of the compressibility constraint and continuity equation, an implicitly higher-order approximation of the velocity field, and a reduced computational domain These advantages have allowed the analysis of a wake region extending hundreds of cylinder diameters, and hence enabled the study of wake transition [15] However, calculating the lift and drag experienced by submerged bodies is complicated by the fact that the pressure variable is eliminated in the vorticity formulation In the current paper, a method of recovering pressure data on the surface of submerged bodies is discussed Results for the nondimensional shedding frequency (Strouhal number), lift, and drag are presented for flow over a cylinder at a Reynolds number of Numerical formulation The numerical formulation is divided into flow kinetics and flow kinematics The problem kinetics are governed by the vorticity equation which is given in two dimensions by ad ~ + (ii V)ti = vv2d, (1) where ii is the velocity field, d = V x ii is the vorticity field, t is time, and u is the constant kinematic fluid viscosity In the course of solving Eq 1, the velocity field, Z, must be determined from the vorticity field, G, and the creation of vorticity on the boundary must be determined from the velocity boundary conditions In the present formulation, determining both the interior velocity field and the creation of vorticity on the boundary are accomplished in a unified manner using the generalized Helmholtz decomposition (GHD) The GHD represents the

3 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) Applications of High-Performance Computing in Engineering VII 107 problem kinematics which is given in two dimensions by (2) where ii is the unit normal vector on the boundary (pointing away from the fluid), Q represents the two-dimensional domain, r is the boundary of Q, and ~ represents the vortex sheet strengths along the boundary The coefficient a is a function of the location of the field point i? For field points outside of the domain, a = O; for field points in the interior of the domain, a = 2m; for field points on smooth portions of the boundary, a = n At edges or corners, CYcan be related to a local internal angle The vortex sheet strengths ~ represent the creation of vorticity on the boundary during a discrete time interval and can be related to the vorticity flux along the boundary using the equation [6] m ~ %=x where At is the discrete time step used in the finite element solution of the vorticity equation The numerical algorithm for solving the vorticity equation is comprised of 2 kinematic and 1 kinetic steps In the first kinematic step, a Galerkin implementation of the GHD is solved to determine vortex sheet strengths These vortex sheet strengths provide Neumann data (Eq, 3) for the vorticity equation In the second kinematic step, the interior velocity field is evaluated using the non-galerkin form of the GHD This, in a sense, linearizes the convective acceleration terms in the vorticity equation In the third kinetic step, a Galerkin finite element method (FEM) is used to solve the vorticity equation After an explicit time step in solving the vorticity equation, kinematic compatibility between the vorticity and velocity field is lost To reestablish kinematic compatibility, the Galerkin GHD is solved again for the vortex sheet strengths which represents vorticity formation during the discrete time step which initiates the next time step Details of this algorithm have been given previously in [6] (3) 3 Evaluation of the surface pressure, lift and drag One of the main purposes of the current research is to demonstrate that, even though pressure is eliminated in the vorticity equation, surface pres-

4 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) 108 Applications of High-Per forntance Computing in Engineering VII sures on a submerged body can be easily computed in postprocessing as discussed below The momentum equation can be written in primitive variables as Dd -Vp-pvxu (4) E = Since the velocity vector Z7is zero on the surface of a submerged body, the normal and tangential components of the momentum equation on the body can be written as =p% ap Ow ~= p~ ap au (5) dt where d/dt and t3/8n represent differentiation in the tangential and normal directions, respectively The drag and lift experienced by a submerged body can be determined by integrating the horizontal and vertical components of the normal (pressure) and shear stresses over the surface of the body The normal stresses can be determined using Eq 5 The shear stress o is determined by au p% where u is the tangential component of the velocity The lift and drag on the submerged body is then determined by Fd = (p + r7)dr FJ= ~(P, + ddr (7) Jr / where pz and pv represent the components of pressure in the x and g directions, respectively and Oz and au represent the components of shear stress in the x and y directions, respectively (6) 4 Parallelization of the vorticity formulation The parallelization of the vorticity formulation is completed in three steps, each of which corresponds directly to one of the three steps of the algorithmic formulation The parallelization of step 1 involves the Galerkln GHD which is essentially a boundary integral equation Assembly of Galerkin boundary integral equations typically involves a double-nested do-loop where both the outer and inner loops are over boundary elements A block-cyclic data distribution results when both the outer and inner loops are distributed over processors This approach is chosen in the current research to minimize communication during the solution phase of the algorithm [5] ScaLAPACK is chosen as the parallel direct solver The matrix is assembled and decomposed in parallel outside of the time loop, so that the only work performed inside the time loop is the updating of the right hand side vector, and the back-substitution to determine the updated values of the vortex sheet

5 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) Applications of High-Performance Computing in Engineering VII 109 strengths The only communication required during the assembly phase are two row-gathersto essentially incorporate the left-hand-side of the discretized GHD into the right-hand-side and to regularize the Cauchy principal value integral However, significant communication is required of the parallel decomposition routine used in ScaLAPACK Within the time loop, additional broadcasts are required to obtain updated values of the vorticity to evaluate the GHD, but overall storage requirements on a single processor are reduced because of the distribution of data among the processors The parallelization of step 2 (evaluation of the velocity integrals) is essentially embarrassingly parallel since each velocity evaluation at the finite element nodes (field points) is independent of all other nodes Hence, the finite element nodes are simply evenly distributed among the processors Outside of the time loop, the integral arrays are determined using Gaussian quadrature Inside the time loop, the velocities are determined on the local processors by performing a simple matrix-vector multiplication The communication required in this step consists of a broadcast of the updated vortex sheet strengths from step 1 and updated vorticities from step 3 and a gather of the velocities for use in step 3 The parallel strategy chosen for step 3 (Galerkin FEM) is the totalsummed-row approach with a row-wrap data distribution [11] Typically, this strategy is based on domain partitioning in which nodal data and connectivity for both the interior elements and border elements required to generate a complete row of the discretized matrix equation reside on a given processor However, in the current implementation, since the nodal data is required for the calculation of the interior velocities in step 1, this information is broadcast to all processors outside of the time loop after it is read in from the input file Hence, no domain partitioning is necessary and rows in the matrix equation can simply be assigned evenly to processors Part of the equation assembly can be performed outside of the time loop However, to assemble the portions of the stiffness matrices associated with the convective acceleration terms, updated values of the velocity field determined in step 2 are required Inside the time loop, the completion of the stiffness matrix is performed by matrix-vector multiplication The load vector is also determined inside the time loop again by performing a matrix-vector multiplication Since the assembly of each row of the matrix equation is independent, there is no communication in the assembly phase of the algorithm The iterative equation solver PIM was chosen to solve the sparse linear system using a restarted Generalized Minimum Residual (GMRES) method with Jacobi preconditioning The data is stored in CSR (compressed sparse row) format and only nonzero matrix values are stored PIM requires a user defined parallel matrix-vector multiplication scheme The majority of the communication required during the solution phase of this step is a gather of the residual vector,

6 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) 110 Applications of High-Performance Computing in Engineering VII 5 Results for flow about a cylinder A free stream of unit velocity is impulsively started at time t = Ofor the problem of external flow about a cylinder of unit diameter The viscosity of the fluid is chosen to be v = 0,005 so that the Reynolds number is given by $2= 200 Results for the Strouhal number, lift and drag generated by the current vorticity formulation are presented below The pressure distribution around the cylinder at various times as calculated in postprocessing of the vorticity code is shown in Fig 1 As seen in the figure, the pressure fluctuates because of the vortex shedding Plots of the coefficient of drag and lift as a function of time are shown in Figs, 2 and 3, respectively Approximately, 85% of the drag and 94% of the lift can be attributable to the pressure forces, The period for the drag is essentially 1/2 the period of the lift A Strouhal number of 0,191 was calculated by taking a fast Fourier transform of the drag experienced by the cylinder A comparison of the current results with results available in the literature is shown in Table 1, The current results are well within the experimental and numerical scatter 0,20 000,1,1,01t=53sec -- - t= S4sec -020 t=55sec t = 56sec t=57sec p ~ -060 : , , ( Angle Around Cylinder (degrees) Figure 1: Pressure distribution on the surface of the cylinder at selected times (Y?= 200)

7 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) Applications of High-Performance Computing in Engineering VII 111 1, A P ~ - /7,f\~\f {l{\~ ) q 10000!! g \ r = - c i j 3 - l\ /F &i08000 r ml w< t,([/,14ji I VIV L1 JP g ++ 4 m ; i - / - ~ Cdp 0 B 06000,,,!,,!,,,,,!,!! Cdv E = al ~ g u E,!,, cd,,,,,,,,,,,,,,,,,,,,,,!,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,000 20, ,000 50, , Time(see) Figure 2: Drag calculations for flow about the cylinder at Y?= 200 CdP is the component of the drag coefficient due to pressure, Cd is the component of the drag coefficient due to viscosity, and Cd is the total drag coefficient 6 Parallel performance The parallel performance of the vorticity code is evaluated using a grid containing 7600 finite elements, 7810 finite element nodes, and 160 boundary elements Log-log plots of the total clock time outside the time loop and per time step inside the time loop are shown in Fig 4 Also shown in the figure is the straight line curve representing ideal scalability Both plots show very good scalability Part of the reason for this is that, typically, over 95% of the total CPU effort is spent in evaluating the velocity field in step 2 This step is the embarrassingly parallel part of the algorithm Nevertheless, step 1 and step 3 also benefit from parallelism not only in reduced clock time, but equally importantly, from reduced storage requirements 7 Conclusions A parallel vorticity code has been developed to analyze flows about prismatic bluff bodies The vorticity formulation has several advantages for this class of incompressible flows However, since the pressure is eliminated in the

8 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) ~ ~2 Applications of High-Performance Computing in Engineering VII , Clp,,,!,,!$!,!,C[v h i I ICL Jh m & : ~ a H u w? H ,000 20, Time (see) Figure 3: Lift calculations for flow about the cylinder at R = 200 CIP is the component of the lift coefficient due to pressure, Clv is the component of the lift coefficient due to viscosity, and Cl is the total lift coefficient vorticit y formulation, a method of retrieving surface pressures has been developed so that lift and drag information can be determined Calculations using the current approach for the coefficient of lift and drag for flows about a cylinder at a Reynolds number of 200 matched very well with results available in the literature The numerical algorithm is divided into three distinct steps, 2 kinematic steps based on the generalized Helmholtz decomposition and 1 kinetic step Each step required a distinct data distribution and parallelization effort The scalability of the overall algorithm proved to be excellent 8 Acknowledgment This work was supported by Sandia National Laboratories Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE- AC04-94-AL85000

9 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) Applications of High-Performance Computing in Engineering VII 113 Current Results Published Values Strouhal Number Drag Coefficient Lift Coefficient Table 1: A comparison of the current results for the Strouhal number, drag coefficient and lift coefficient with results available in the literature [3, 4,7, 10, 12, 16] g \ Ico 00 l\ \ 80 OutW@ TIMLCCQ \ - -A- - I*1 scaling \ Qw 70 E a \ a! Number of Processors Number of Processors Figure 4: Clock times (a) outside the time loop and (b) inside the time loop per time step for the flow about the cylinder problem References [1] E1-Refaee, M M, Boundary layer control of separated flow over circular cylinders - a BEM parametric study, Eng Anal, BEM, 14(3), , 1994 [2] Engelman, M S, Jamnia, M-A, Transient flow past a circular cylinder: A benchmark solution, Int, J Num Meth Fluids, 11, , 1990 [3] Farrant, T, Tan, M, and Price, W G, A cell boundary element method applied to laminar vortex shedding from circular cylinders, Comput & Fluids, 30, ,2001 [4] Gresho, P M, Chan S T,, Lee, R L, and Upson, C D, A modified finite element method for solving time-dependent, incompressible Navier- Stokes equations Part 2: Applications, Int, J Num Meth Fluids, 4, , 1984

10 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK All rights reserved Web: wwwwitpresscom Paper from: Applications of High Performance Computing in Engineering VII, CA Brebbia, P Melli & A Zanasi (Editors) ~ 14 Applications of High-Performance Computing in Engineering VII [5] Hendrickson, B A, and Womble, D E, The torus-wrap mapping for dense matrix calculations on massively parallel computers, SIAM J Sci Comput, 15(5), ,1994 [6] Ingber, M, S and Kempka, S N, A Galerkin implementation of the generalized Helmholtz decomposition for vorticity formulations, J Comp Ph~s,, 169, ,2001, [7] Inoue, O, Yamazaki, T, and Bisaka, T,, Numerical simulation of forced wakes around a cylinder, Int J Heat Fluid Flow, 16, , 1995 [8] KArm6n, T Von, Uber den mechanisms des wilderstandes etc, 1, Teil, Nachr Wiss Ges, G6ttingen, Math Phys, Kl,, , 1911 [9] Li, J, Chambarel, A,, Donneaud, M and Martin, R, Numerical study of laminar flow past one and two cylinders, Fluids, 19(2), , 1991 [10] Lu, X-Y, and Dalton, C Calculation of the timing of vortex formation from an oscillating cylinder, J Fluids Struct, 10, , 1996 [11] Shadid, J N, Hutchinson, S A, Moffat, H K, Henningan, G L, and Hendrickson, B, A 65+ Gflop/s Unstructured Finite Element simulation of chemically reacting flows on the Intel Paragon, in Proceedings of Supercomputing 94, IEEE Computer Society Press, 1994 [12] Tamura, T, Ohta, I, and Kuwahara, K,, on the reliability of twodimensional simulation for unsteady flows around a cylinder-type structures, J Wind Engrg, Ind Aero, 35, , 1990 [13] Thorn, A, The flow past circular cylinders at low speeds, Royal Sot, A141, , 1933 [14] Tritton, D, J, A note on vortex streets behind circular cylinders at low Reynolds numbers, J Fluid Mech, 45, ,1959 [15] Vorobieff, P, Georgiev, D, and Ingber, M S, onset of the second wake: Dependence on the Reynolds number, to appear Phys, Fluids, 2002 [16] Williamson, C H K and Roshko, A, Measurements of base pressure in the wake of a cylinder at low reynolds numbers, Zeit fiir Flug Welt, 14, 38-46, 1990