This paper presents the first high-order computational fluid dynamics (CFD) simulations of static and spinning golf balls at realistic flow conditions. The present results are shown to capture the complex fluid dynamics inside the dimples which lead to drag reduction versus a smooth sphere, and compare well to previous experimental and computational studies. The high-order flux reconstruction method has been paired with the artificial boundary overset method to enable simplified mesh generation and grid motion. The compressible Navier–Stokes equations are modeled using a scale-resolving large eddy simulation (LES) approach with no sub-grid models. The codes implementing these methods have been implemented for NVIDIA graphical processing units (GPUs), enabling large speedups over traditional computer hardware. The new method allows for the simulation of golf balls, and other objects at moderate Reynolds numbers, to be simulated in a matter of days on large computing clusters.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aoki K, Muto K, Okanaga H (2010) Aerodynamic characteristics and flow pattern of a golf ball with rotation. Procedia Eng 2:2431–2436
Asthana K, Jameson A (2014) High-order flux reconstruction schemes with minimal dispersion and dissipation. J Sci Comput 62:913–944
Bearman PW, Harvey JK (1976) Golf ball aerodynamics. Aeronaut Q 27:112–122
Beratlis N, Squires K, Belaras E (2012) Numerical investigation of Magnus effect on dimpled spheres. J Turbul 13(15):1–15. https://doi.org/10.1080/14685248.2012.676182
Butcher JC (2016) Numerical methods for ordinary differential equations, 3rd edn. Wiley, New York
Castonguay P (2012) High-order energy stable flux reconstruction schemes for fluid flow simulations on unstructured grids. Ph.D. thesis, Stanford University
Choi J, Jeon WP, Choi H (2006) Mechanism of drag reduction by dimples on a sphere. Phys Fluids 18:041702
Chowdhury H, Loganathan B, Wang Y, Mustary I, Alam F (2016) A study of dimple characteristics on golf ball drag. Procedia Eng 147:87–91
Crabill J, Jameson A, Sitaraman J (2016) A high-order overset method on moving and deforming grids. In: AIAA modeling and simulation technologies conference (AIAA2016-3225). https://doi.org/10.25146/2016-3225
Crabill JA (2018) Towards industry-ready high-order overset methods on modern hardware. Ph.D. thesis, Stanford University
Crabill JA, Witherden FD, Jameson A (2018) A parallel direct cut algorithm for high-order overset methods with application to a spinning golf ball. J Comput Phys 374:692–723
Eiseman PR, Ebenezer SJ, Sudharshanam V, Anbumani V (2016) GridPro manuals, version 2.0. http://www.gridpro.com. Accessed 2 Sept 2016
Galbraith MC (2013) A discontinuous galerkin overset solver. Ph.D. thesis, University of Cincinatti
Geuzaine C, Remacle JF (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79:1309–1331 (Mesh-Generation Software)
Hindenlang F, Bolemann T, Munz CD (2015) Mesh curving techniques for high order discontinuous Galerkin simulations. In: Kroll N, Hirsch C, Bassi F, Johnston C, Hillewaert K (eds) IDIHOM: industrialization of high-order methods—a top–down approach. Springer, Berlin, pp 133–152
Huynh HT (2007) A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 47th AIAA aerospace sciences meeting
Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392
Kennedy CA, Carpenter MH, Lewis RM (2000) Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl Numer Math 35(3):177–219
Langguth J, Wu N, Cai JCX (2013) On the GPU performance of cell-centered finite volume method over unstructured tetrahedral meshes. In: S. SC13 International conference for high performance computing networking, analysis (eds.) \(IA^3\): Workshop on irregular applications: architectures and algorithms. ACM, Denver, CO, USA. https://doi.org/10.1145/2535753.2535765
Li J, Tsubokura M, Tsunoda M (2015) Numerical investigation of the flow around a golf ball at around the critical Reynolds number and its comparison with a smooth sphere. Flow Turbul Combust 95:415–436. https://doi.org/10.1007/s10494-015-9630-4
Li J, Tsubokura M, Tsunoda M (2017) Numerical investigation of the flow past a rotating golf ball and its comparison with a rotating smooth sphere. Flow Turbul Combust 99:837–864. https://doi.org/10.1007/s10494-017-9859-1
Mehta RD (1985) Aerodynamics of sports balls. Ann Rev Fluid Mech 17:151–189
Muto M, Tsubokura M, Oshima N (2012) Negative Magnus lift on a rotating sphere at around the critical Reynolds number. Phys Fluids 24:014102
Romero JD (2017) On the development of the direct flux reconstruction scheme for high-order fluid flow simulations. Ph.D. thesis, Stanford University
Sitaraman J (2015) TIOGA: topology independent overset grid assembly library. https://github.com/jsitaraman/tioga (Overset Connectivity Library for CFD). Accessed 26 June 2018
Smith CE, Beratlis N, Balaras E, Squires K, Tsunoda M (2010) Numerical investigation of the flow over a golf ball in the subcritical and supercritical regimes. Int J Heat Fluid Flow 31:262–273. https://doi.org/10.1016/j.ijheatfluidflow.2010.01.002
Smits AJ, Smith DR (1994) A new aerodynamic model of a golf ball in flight. In: Cochran AJ, Farrally MR (eds) Science and golf II: proceedings of the world scientific congress of golf. E & FN Spon, London, pp 340–347
Ting LL (2002) Application of CFD technology analyzing the three-dimensional aerodynamic behavior of dimpled golf balls. In: ASME international mechanical engineering congress & exposition
Ting LL (2003) Effects of dimple size and depth on golf ball aerodynamics. In: 4th ASME/JSME joint fluids engineering conference
Vermeire BC, Witherden FD, Vincent PE (2017) On the utility of GPU accelerated high-order methods for unsteady flow simulations: a comparison with industry-standard tools. J Comput phys 334:497–521
Vincent P, Castonguay P, Jameson A (2010) A new class of high-order energy stable flux reconstruction schemes. J Sci Comput 47:50–72
Vincent P, Castonguay P, Jameson A (2011) Insights from von Neumann analysis of high-order flux reconstruction schemes. J Comput phys 230:8134–8154
Vincent P, Witherden F, Vermeire B, Park JS, Iyer A (2016) Towards green aviation with Python at petascale. In: Proceedings of the international conference for high performance computing, networking, storage and analysis, SC ’16. IEEE Press, Piscataway, NJ, USA. http://dl.acm.org/citation.cfm?id=3014904.3014906
Williams DM, Castonguay P, Vincent PE, Jameson A (2013) Energy stable flux reconstruction schemes for advection–diffusion problems on triangles. J Comput Phys 230:8134–8154
Wissink A (2012) Helios solver developments including strand meshes. In: Oral presentation, 11th symposium on overset composite grids and solution technology
Wissink A (2012) An overset dual-mesh solver for computational fluid dynamics. In: 7th International conference on computational fluid dynamics (ICCFD7)
Witherden FD, Farrington AM, Vincent PE (2014) PyFR: an open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach. Comput Phys Commun 185:3028–3040
Witherden FD, Vincent PE, Jameson A (2016) High-order flux reconstruction schemes. In: Handbook of numerical analysis, vol 17, pp 227–2633. Elsevier, Boca Raton. https://doi.org/10.1016/bs.hna.2016.09.010 (chap. 10)
The authors would like to acknowledge the Army Aviation Development Directorate (AMRDEC) for providing funding for this research under the oversight of Roger Strawn, the Air Force Office of Scientific Research for their support under Grant FA9550-14-1-0186 under the oversight of Jean-Luc Cambier, and Margot Gerritsen for access to the XStream GPU computing cluster, which is supported by the National Science Foundation Major Research Instrumentation program (ACI-1429830). We would also like to thank Dr. Peter Eiseman for providing academic licensing to the GridPro meshing software and assisting with the creation of several golf ball grids. Last, we would like to thank Dr. Jay Sitaraman for his expertise and help on overset connectivity methods, and his help in ensuring our numerical methods were robust enough for broad applicability.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Crabill, J., Witherden, F. & Jameson, A. High-order computational fluid dynamics simulations of a spinning golf ball. Sports Eng 22, 9 (2019). https://doi.org/10.1007/s12283-019-0300-y
- Computational fluid dynamics
- Large eddy simulation
- Finite element methods
- Golf ball
- Sports aerodynamics