Analytical approximation of solutions to the Blasius problem in the boundary layer on a flat plate
https://doi.org/10.26907/2541-7746.2025.2.227-243
Abstract
In recent years, the steady-state flow of viscous fluids has attracted considerable research interest due to its broad engineering applications. This study provides new insights into the classical problem of the theory of the viscous laminar stationary boundary layer of an incompressible Newtonian fluid on a thin flat plate (Blasius problem). A finite-difference solution to the Blasius problem was obtained by the shooting method in combination with the Runge–Kutta numerical scheme of the fourth-order accuracy over a large interval for a very fine mesh. The numerical results were validated against similar known data of calculation tests. The Blasius function f(η) and its first two derivatives were approximated using the third-order B-spline. Excellent agreement with the results of known calculations was demonstrated. A new analytical correlation for the Blasius function, which approximates the results of the calculations in a wide range of the self-similar variable η, was established by the nonlinear least squares method (NLLSM). The values of the function f and its first- and second-order derivatives were compared with known data. The results align with previous solutions. The longitudinal velocity profile in the boundary layer, defined through the derivative f′ of the Blasius function, can serve as the initial velocity profile in the numerical modeling of turbulent flat and three-dimensional flows of an incompressible fluid.
Keywords
laminar boundary layer,
steady-state flow,
incompressible Newtonian fluid,
thin flat plate,
Blasius problem,
B–spline,
analytical approximation,
nonlinear least squares method (NLLSM)
Supporting agencies: This study was performed as part of the state assignment (state registration no. 124012500440-9)
About the Author
V. M. Zubarev
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
Vyacheslav M. Zubarev, Cand. Sci. (Physics and Mathematics), Researcher
Moscow
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For citations:
Zubarev V.M. Analytical approximation of solutions to the Blasius problem in the boundary layer on a flat plate. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(2):227-243. (In Russ.) https://doi.org/10.26907/2541-7746.2025.2.227-243
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