EULER AND SECOND-ORDER RUNGE-KUTTA METHODS FOR COMPUTATION OF FLOW AROUND A CYLINDER

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2 EULER AND SEOND-ORDER RUNGE-KUTTA METHODS FOR OMPUTATION OF FLOW AROUND A YLINDER László Daróczy, László Baranyi MSc student, Professor University of Miskolc, Hungary Department of Fluid and Heat Engineering, University of Miskolc, Hungary ABSTRAT The two-dimensional low-reynolds number flow around a circular cylinder is investigated numerically using two different time discretisation methods ( st order Euler and nd order Runge-Kutta) and two different types of hardware (PU- and GPGPU-based) using the finite difference method. omputations were done at Reynolds number and 5 for different dimensionless time steps (.;.;.4;.5) and different mesh sizes (545; 366). For all results flow properties such as drag, lift and base pressure coefficients are analysed and compared with each other. omputational results obtained using the four different types of code agree well, and are also very similar to values in the literature. INTRODUTION When computing flow around a bluff body such as a cylinder a proper balance needs to be found between computational time and accuracy. Higher order computational methods are more accurate than lower order methods when the temporal and spatial disretisations are the same. If a higher order method is chosen then it is possible to obtain the same accuracy as for a lower order method even at a larger time step and or with a coarser mesh. The Euler ( st order) method, although considered old-fashioned by many, is still in use, even for problems involving highly unsteady phenomenon such as wake flows of bluff bodies. Among the nd order methods the most frequently used are the Runge- Kutta and the Adams-Bashforth methods []; several versions of each eist. There are also 4 th order Runge Kutta methods but they are not so frequently used in the study of wake flow. For this sudy we tried the four most often recommended versions of nd order Runge-Kutta (RK) method: the Heun, Midpoint, Ralston and Taylor series methods [, 3]. st and nd order results are compared at different time steps, grid sizes, computational domains and hardware. The D computations are for a low-reynolds number flow around a stationary circular cylinder placed in a uniform stream. EULER AND RUNGE-KUTTA METHODS We briefly introduce the computational procedure developed by the second author [4, 5] incorporating the Euler method. A non-inertial system fied to the accelerating cylinder is used to compute two-dimensional (D) low Reynolds

3 number flow around a circular cylinder placed perpendicular to an otherwise uniform flow. The governing equations are the Navier-Stokes equations, the equations of continuity and a Poisson equation for pressure. On the cylinder surface no-slip boundary condition is used for the velocity and a Neumann type boundary condition for the pressure. At the far region potential flow is assumed. Boundaryfitted coordinates are used to impose boundary conditions accurately. The physical domain, which consists of two concentric circles (with radii R, R ), is mapped into a rectangular computational domain. The governing equations and boundary conditions are also transformed to the computational domain (not shown here) and solved by finite difference method. Space derivatives are approimated by fourthorder central differences, ecept for the convective terms, for which a third-order modified upwind scheme is used. The Poisson equation for pressure is solved by the successive over-relaation (SOR) method. The Navier-Stokes equations are integrated using the Euler method to yield the velocity field. The continuity equation is satisfied at every time step. For further details see [4] and [5]. We tested all four Runge-Kutta methods mentioned, but finally Heun's method was chosen for implementation as a compromise between speed and accuracy. The dimensionless Navier-Stokes equation for a non-inertial system fied to the accelerating cylinder is v = ( v ) v p+ v a, () t Re where v is the velocity vector, p is the pressure, Re = Ud / ν is the Reynolds number based on the free stream velocity U, cylinder diameter d and kinematic viscosity of the fluid ν, a is the cylinder acceleration vector ( here), and t is the dimensionless time. The pressure Poisson equation can be written as [4] D u v u v = p t y y. () With superscript n denoting a time level and t a time step, the sequence of temporal integration of equation () using Heun s RK method is as follows. Step. Determine RK coefficient k and intermediate velocity v and pressure p n n n n k= ( v ) v + p v Re. (3) Using k and equation () the intermediate velocity v can be obtained v = v + k t v + v, (4) n n n + where t is the time step and v is the velocity of the cylinder. Now we can determine an intermediate pressure p using v and equation ()

4 n D u v u v = p t y y, (5) n n where D = div( u ). In this way while solving the Poisson equation (5) we also enforce the continuity condition of D = div( v ) =. n+ n Step. Determine RK coefficient k and final velocity u and pressure p + k = ( v ) v + p v Re, (6) n n n n v + = v + ( k+ k ) t v + + v, (7) n n+ n+ n+ n+ D u v u v n+ = p t y y. (8) As can be seen from this sequence, the SOR method, which requires the large bulk of the computational time, has to be applied twice in a single time step when using the Runge-Kutta method, while with the Euler method it is applied only once. However, tests showed that in most cases the computational time does not double. DIFFERENT TYPES OF ODES In the current study FlowFD software, developed by the first author, provided the framework for Euler and Runge-Kutta methods, running on either PU or GPU. The software package includes many post-processing options, including visualization of flow and limit cycles in D and 3D, analysis of different coefficients, and management for scheduling simulations. GPU codes use a GPGPU (General-purpose computing on graphics processing units) programming method. The code itself is ++ based, while the computation kernels are written in NVIDIA UDA. Instead of the original SOR method [4], the GPU version uses a red-black Gauss-Seidel iteration with successive overrelaation (SOR) [6]. The results presented in the present article were all run on Tesla 5 using 448 cores with double precision and E on. The study uses four different variations of FD codes. Types and are the recreation in ++ of the original FORTRAN code developed by the second author [4], using st order Euler-type time discretisation; Type uses the PU, and Type runs on the GPU. Types 3 and 4 are based on the same code, slightly modified for the Runge-Kutta method; Type 3 runs on the PU and Type 4 on the GPU. TESTED ASES & EVALUATION METHODS and 5 cases were tested for a stationary cylinder. For the GPU codes a 545 (peripheral radial) grid was used at both Reynolds numbers, and for PU codes two grids were used: for 5 a 545 grid with R / R = 6, while the same was used for, in addition to a smaller grid of 366 ( R / R = 6).

5 For all cases 4 4 t = was used with ( ) t=,,4,5 non-dimensional time step. 4 For Reynolds number time step t 6 failed to converge. As Fast Fourier Transform (FFT) analysis revealed only one dominant frequency in the spectrum, for evaluation of the results the built-in Maut algorithm was used, which calculates the period time of the oscillations using the local maima of the coefficients. The vorticity contours were the same for all cases belonging to the same Reynolds number (see Figure ). RESULTS Figure Vorticity contours () The results are presented in Tables and for the different settings. Drag coefficient ( D, mean), lift coefficients (amplitude of L ) and base pressure coefficients (, mean) are given. The results are compared to the values pb presented in the work of Posdziech and Grundmann, who used a systematic approach to determine force coefficients and Strouhal number of two-dimensional flow around a circular cylinder using the spectral element method [7]. We can see that the results correlate well: for eample, values are presented in [7] as.33 5 Hardware D Table Results - PU Time step Euler (st order) Runge-Kutta (nd order) Δt (dimensionless) PU D pb L D pb L D pb L

6 (for their largest domain size, at ), which is similar to the range of values presented in Tables and ( for both PU and GPU). They also gave a value of.99 for 5 (around.3 for us), and values are D presented as.69 (for the largest domain size, ) and.87 for 5. Hardware Table Results - GPU GPU Time step Euler (st order) Runge-Kutta (nd order) Δt (dimensionless) D pb L D pb L The non-dimensional vorte shedding frequency St = fd / U is also compared with the results in [7]. In case of, for comparison we will use the value of.633 ([7], Table 4, largest domain). The values were in the range of , with relative errors in the range of.576%.636%. The Runge- Kutta codes resulted in smaller relative error. However, for the smaller grid (366) we obtained results in the range of , corresponding to the relative error of.69%.9%, slightly higher than in the case of the larger grid. For 5, [7] suggests several formulas to evaluate St for different Reynolds numbers. Using the non-integer polynomial (lowest error) we get omparing these values with the tested cases, we see that results are between.8356 and.8365, which is a very low error (between.6% and.58%). For L amplitudes the presented values in [7] are.36 (, largest domain size) and.53 (5, largest domain). The amplitude of the lift coefficient compared well with the values in [7]. In the case of, for the smaller mesh values were in the range of.35.34, while for the larger mesh they were For 5, our simulations gave values in the range of ONLUSIONS The second order results agree very well with the results of the first order temporal discretisation for the time steps investigated. This indicates that the first order results are accurate enough for practical cases under the conditions investigated. Both sets of results agree well with the spectral element results given in [7]. In addition, performing the computations using PU and GPGPU revealed no pb

7 significant differences in the results, although the computational time required is roughly 8- times lower for the GPU. While we were able to implement the Runge-Kutta method into our eisting and proven code we found that we were unable to take advantage of the benefits of higher order methods, i.e., being able to use larger time steps and still obtain the same accuracy. When the time step was increased the SOR method failed to converge. It appears that the SOR method and the Runge-Kutta method are somewhat incompatible. We are currently investigating ways to replace SOR. The PU Euler code is the identical to the original FORTRAN version, which has been used many times in publications and articles, and whose results correlate well with the literature [5]. From these results it can be clearly concluded that the GPGPU and PU-based code yield identical results up to the defined precision 5 ( ). omparing the results of GPU codes we can see the same tendency. AKNOWLEDGEMENTS The authors would like to say special thanks to NVIDIA for donating an NVIDIA Tesla 5 within the Academic Partnership Program. The support provided by the Hungarian Research Foundation (OTKA project K 768) is gratefully acknowledged. The work was carried out as part of the TÁMOP-4...B- //KONV-- project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union, co-financed by the European Social Fund. REFERENES [] FERZIGER, J.H., PERIĆ, M.: omputational Methods for Fluid Dynamics (3rd edition). Springer-Verlag, Berlin (). [] HEEVER, E.: The nd order Runge-Kutta method. [3] KAW, A.K., KALU, E.E.: Numerical Methods with Applications. Ed. World Wide Web, 9. [4] BARANYI, L.: omputation of unsteady momentum and heat transfer from a fied circular cylinder in laminar flow. Journal of omputational and Applied Mechanics 4() (3), 3-5. [5] BARANYI, L.: Numerical simulation of flow around an orbiting cylinder at different ellipticity values. Journal of Fluids and Structures 4 (8), [6] SHINN, A.F.: omputational Fluid Dynamics (FD) using Graphics Processing Units. Accelerators for Science and Engineering Applications: GPUs and Multicores [7] POSDZIEH, O., GRUNDMANN, R.: A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder. Journal of Fluids and Structures 3 (7),

Numerical Simulation of Mechanical Energy Transfer between Fluid and a Circular Cylinder Forced to Follow an Elliptical Path

OS16-3-3 The 12th International Symposium on Fluid Control, Measurement and Visualization November183, 213, Nara, Japan FLUCOM213 Numerical Simulation of Mechanical nergy Transfer between Fluid and a Circular