New explicit solution of the Blasius equation in the boundary layer around the hull of a ship by approximation of derivatives

Document Type : Research Paper

Authors

1 Laboratoire de Mécanique d’Ingénierie et d’Innovation LM2I, ENSEM, Université Hassan II, Casablanca 20103, Morocco.

2 Département Energie, Ecole Royale Navale, Casablanca 20052, Morocco.

Abstract

Naval hydrodynamics fundamentally depends on a detailed understanding of the boundary layers forming around a ship’s hull, which generate resistance to advancement. Accurately modeling these layers is critical for calculating hydrodynamic resistance and estimating the propulsion power needed to achieve the desired speed specified by the shipowner. Traditionally, the velocity distribution within the boundary layer is described by the Blasius equation, a nonlinear third-order differential equation commonly solved using the Runge-Kutta numerical method, renowned for its accuracy.
This study proposes a novel direct and explicit approach to solving the Blasius equation around a ship’s hull, leveraging a derivative approximation technique implemented with MATLAB to obtain numerical results. By employing sufficiently small step sizes, the method produces highly accurate results that can serve as a benchmark for evaluating the precision of other numerical techniques applied in ship design. The proposed derivative approximation method provides a simple yet robust tool for solving complex differential equations, demonstrating its potential as an effective alternative for tackling problems similar to the Blasius equation in naval engineering applications.

Keywords

Main Subjects

References

  • [1] A. Aghajani and H. Mirafzal, Numerical Simulation of Boundary Layer Flow with Variable Viscosity and ThermoPhysical Properties, Computational Methods for Differential Equations, 9 (2021), 325–338.
  • [2] M. Ahmed and M. Yousuf, Numerical Solution of Nonlinear Boundary Layer Equations for Variable Viscosity Fluids, Journal of Computational and Applied Mathematics, 378 (2020), Article ID 112637.
  • [3] N. S. Asaithambi, A numerical method for the solution of the Falkner-Skan equation, Applied Mathematics and Computation, 81 (1997), 259–264.
  • [4] A. Asaithambi, Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients, Journal of Computational and Applied Mathematics, 176 (2005), 203–214.
  • [5] R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier Publishing Company, New York, 1965.
  • [6] H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Zeitschrift fu¨r angewandte Mathematik und Physik, 56 (1908), 1–37.
  • [7] T. Cebeci and H. Keller, Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equations, Journal of Computational Physics, 7 (1971), 289–300.
  • [8] R. Cortell, Numerical solutions of the classical Blasius flat-plate problem, Applied Mathematics and Computation, 170 (2005), 706–710.
  • [9] G. Chen and et al., Analysis of Viscosity Variations in Non-Isothermal Blasius Boundary Layers, Computational Methods for Differential Equations, 10 (2022), 245–258.
  • [10] V. M. Falkner and S. W. Skan, LXXXV. solutions of the boundary-layer equations, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 12 (1931), 865–896.
  • [11] T. Fang, W. Liang, and C. F. Lee, A new solution branch for the Blasius equation – A shrinking sheet problem, Computers & Mathematics with Applications, 56 (2008), 3088–3095.
  • [12] T. Fang and J. Zhang, An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching, International Journal of Non-Linear Mechanics, 43 (2008), 1000–1006.
  • [13] R. Fazio, The iterative transformation method for the Sakiadis problem, Computers & Fluids, 106(Supplement C) (2015), 196–200.
  • [14] D. Georgieva and V. M. Grahovski, An Efficient Numerical Approach for Solving the Blasius Equation for Boundary Layers in Nanofluid Flow, Computational Methods for Differential Equations, 11 (2023), 117–125.
  • [15] M. Goody, T. Farabee, and Y. Lee, Unsteady Pressures on the Surface of a Ship Hull, in Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition, vol. 1, Advances in Aerospace Technology, 2007, 79–86, Seattle, Washington, USA, November 11–15.
  • [16] S. Gupta and R. Singh, Numerical Methods for Solving Boundary Layer Problems with Variable Thermo-Physical Properties, Computational and Applied Mathematics, 42 (2023), 157–172.
  • [17] D. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer, Mathematical Proceedings of the Cambridge Philosophical Society, 33 (1937), 223–239.
  • [18] Z. Harun, J. P. Monty, R. Mathis, and I. Marusic, Pressure gradient effects on the large-scale structure of turbulent boundary layer, Journal of Fluid Mechanics, 715 (2013), 477–498.
  • [19] K. Hiemenz, Die Grenzschicht an einem in den gleichf¤ormigen Fl¤u¨ssigkeitsstrom eingetauchten geraden Kreiszylinder, Polytechnisches Journal, 326 (1911), 391–393.
  • [20] J. Holtrop and G. G. J. Mennen, An approximate power prediction method, International Shipbuilding Progress, 29 (1982), 166–170.
  • [21] F. Homann, Der Einfluss grosser Z¤ahigkeit bei der Stromung um den Zylinder und um die Kugel, Zeitschrift fur Angewandte Mathematik und Mechanik, 16 (1936), 153–164.
  • [22] L. Howarth, On the solution of the laminar boundary layer equations, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 164 (1938), 547–579.
  • [23] S. H. Hwang, H. S. Ahn, Y. Y. Lee, M. S. Kim, S. H. Van, K. S. Kim, J. Kim, and Y. H. Jang, Experimental study on the bow hull-form modification for added resistance reduction in waves of KVLCC2, in Proceedings of the 26th International Ocean and Polar Engineering Conference, 2016, 864–868, Rhodes, Greece.
  • [24] R. Jafari, Computational Analysis of Thermo-Physical Properties in Boundary Layers with Variable Viscosity, Advances in Mathematical Physics, 2021, Article ID 3345228.
  • [25] M. Khan, T. Salahuddin, and Others, A Blasius boundary layer analysis for variable viscosity function near a flat plate, International Communications in Heat and Mass Transfer, 135 (2022), Article 106320.
  • [26] M. Khan and T. Salahuddin, A Blasius boundary layer study for generalized viscosity model with thermo-physical properties and Falkner-Skan approach, Alexandria Engineering Journal, 81 (2023), 444-448..
  • [27] J. Kim, H. Park, and M. Lee, Falkner-Skan Flow with Variable Viscosity: A Numerical Study, Computational Mathematics and Modeling, 31 (2020), 291–305.
  • [28] S. Kumar, Boundary Layer Analysis Using the Falkner-Skan Model: A Comparative Study of Numerical Methods, Computational Methods for Differential Equations, 8 (2020), 280–294.
  • [29] L. Kumar, M. Shah, and R. Prasad, Soret and Dufour Effects on MHD Boundary Layer Flow, Computational Methods for Differential Equations, 11 (2023), 45–57.
  • [30] A. Lahlali, Z. El Maskaoui, L. Bousshine, and A. Dinane, Development of a New Concept of Multimission Pollution Control Vessel and Optimization of its Bulbous Bow, Journal of Marine Science and Application, 22 (2023), 513– 526.
  • [31] H. Lee and J. Lim, Laminar and Turbulent Transition in Boundary Layers: A Numerical Investigation Based on Modified Blasius Equations, Computational Methods for Differential Equations, 9 (2021), 399–412.
  • [32] Q. Liang and S. Wei, Blasius Boundary Layer Solutions for Power-Law Fluids, Applied Mathematics and Computation, 376 (2020), Article ID 125210.
  • [33] S. Liao, A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate, Journal of Fluid Mechanics, 385 (1999), 101–128.
  • [34] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall, CRC Press, 2003, Boca Raton.
  • [35] C. S. Liu and J. R. Chang, The Lie-group shooting method for multiple-solutions of Falkner-Skan equation under suction-injection conditions, International Journal of Non-Linear Mechanics, 43 (2008), 844–851.
  • [36] A. F. Molland, S. R. Turnock, and D. A. Hudson, Ship resistance and propulsion: Practical estimation of ship propulsive power, Cambridge University Press, 2011.
  • [37] S. Mukhopadhyay, I. C. Mondal, and A. J. Chamkha, Casson fluid flow and heat transfer past a symmetric wedge, Heat Transfer–Asian Research, 42 (2013), 665–675.
  • [38] A. Papanikolaou, Ship Design. Methodologies of Preliminary Design, Springer, 2014.
  • [39] K. Pohlhausen, Zur naherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht, Journal of Applied Mathematics and Mechanics (ZAMM), 1 (1921), 252–268.
  • [40] S. Patel and A. Dubey, Investigation of MHD Flow Over a Stretching Sheet with Variable Viscosity Using Numerical Methods, International Journal of Computational Fluid Dynamics, 35 (2021), 242–255.
  • [41] C. S. K. Raju and N. Sandeep, Nonlinear radiative magnetohydrodynamic Falkner-Skan flow of Casson fluid over a wedge, Alexandria Engineering Journal, 55 (2016), 2045–2054.
  • [42] C. S. K. Raju, S. M. Ibrahim, S. Anuradha, and P. Priyadharshini, Bio-convection on the nonlinear radiative flow of a Carreau fluid over a moving wedge with suction or injection, The European Physical Journal Plus, 131 (2016), 409–409.
  • [43] C. S. K. Raju, M. M. Hoque, and T. Sivasankar, Radiative flow of Casson fluid over a moving wedge filled with gyrotactic microorganisms, Advanced Powder Technology, 28 (2017), 575–583.
  • [44] C. S. K. Raju and N. Sandeep, MHD slip flow of a dissipative Casson fluid over a moving geometry with heat source/sink: A numerical study, Acta Astronautica, 133 (2017), 436–443.
  • [45] P. L. Sachdev, R. B. Kudenatti, and N. M. Bujurke, Exact analytic solution of a boundary value problem for the Falkner-Skan equation, Studies in Applied Mathematics, 120 (2008), 1–16.
  • [46] Z. Shah and T. Khan, A Numerical Approach to Solve Blasius-Type Equations with Variable Viscosity, Journal of Applied and Computational Mathematics, 12 (2022), 310–322.
  • [47] H. Sadat Hosseini, P. C. Wu, P. M. Carrica, H. Kim, Y. Toda, and F. Stern, CFD verification and validation of added resistance and motions of KVLCC2 with fixed and free surge in short and long head waves, Ocean Engineering, 59 (2013), 240–273.
  • [48] A. A. Salama, Higher-order method for solving free boundary-value problems, Numerical Heat Transfer, Part B: Fundamentals, 45 (2004), 385–394.
  • [49] I. Sher and A. Yakhot, New approach to solution of the Falkner-Skan equation, American Institute of Aeronautics and Astronautics, 39 (2001), 965–967.
  • [50] H. Temimi and M. Ben Romdhane, An iterative finite difference method for solving Bratu’s problem, Journal of Computational and Applied Mathematics, 292 (2016), 76–82.
  • [51] H. Temimi and M. Ben-Romdhane, Numerical Solution of Falkner-Skan Equation by Iterative Transformation Method, Mathematical Modelling and Analysis, 23 (2018), 139–151.
  • [52] A. Vaidya and P. Sharma, Boundary Layer Flow Analysis of Nano Fluids with Variable Viscosity in a Porous Medium, International Journal of Heat and Mass Transfer, 135 (2019), 456–465.
  • [53] D. Xu and X. Guo, Application of fixed point method to obtain semi-analytical solution to Blasius flow and its variation, Applied Mathematics and Computation, 224 (2013), 791–802.
  • [54] H. Yamada, Y. Tanaka, and S. Nishimura, Laminar Boundary Layer Flow in Nanofluids Over a Stretching Sheet, International Journal of Computational Fluid Dynamics, 35 (2021), 215–227.
  • [55] T. Zhang and B. Liu, Generalized Viscosity and its Effect on the Blasius Boundary Layer Flow Near a Flat Plate, Applied Mathematical Modelling, 91 (2022), 788–804.
  • [56] S. Zhu, Q. Wu, and X. Cheng, Numerical solution of the Falkner-Skan equation based on quasi-linearization, Applied Mathematics and Computation, 215 (2009), 2472–2485.
Computational Methods for Differential Equations
Volume 14, Issue 2
April 2026
Pages 780-797

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History

  • Receive Date: 20 June 2024
  • Revise Date: 23 December 2024
  • Accept Date: 17 February 2025

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