Abstract
The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities.
REFERENCES
V. V. Sychev, A. I. Ruban, Vik. V. Sychev, and G. L. Korolev, Asymptotic Theory of Separation Flows, Ed. by V. V. Sychev (Nauka, Moscow, 1987) [in Russian].
F. K. Moore, “On the separation of the unsteady laminar boundary-layer,” in Boundary Layer Research, Ed. by H. Görtler (Springer, Berlin, 1958), pp. 296–311.
N. Rott, “Unsteady viscous flow in the vicinity of a stagnation point,” Q. Appl. Math. 13 (4), 444–451 (1956).
W. R. Sears, “Some recent developments in airfoil theory,” J. Aeronaut. Sci. 23 (5), 490–499 (1956).
S. Timoshin, “Concerning marginal singularities in the boundary-layer flow on a downstream-moving surface,” J. Fluid Mech. 308, 171–194 (1996).
A. I. Ruban, D. Araki, R. Yapalparvi, and J. S. B. Gajjar, “On unsteady boundary-layer separation in supersonic flow. Part 1: Upstream moving separation point,” J. Fluid Mech. 678, 124–155 (2011).
V. I. Zhuk, “On local recirculation zones in the supersonic boundary layer on a moving surface,” USSR Comput. Math. Math. Phys. 22 (5), 249–255 (1982).
R. Yapalparvi and L. Van Dommelen, “Numerical solution of unsteady boundary-layer separation in supersonic flow: Upstream moving wall,” J. Fluid Mech. 706, 413–430 (2012).
A. I. Ruban, A. Djehizian, J. Kirsten, and M. A. Kravtsova, “On quasi-steady boundary-layer separation in supersonic flow: Part 2. Downstream moving separation point,” J. Fluid Mech. 900, A9-1–A9-32 (2020).
S. N. Timoshin and P. Thapa, “On-wall and interior separation in a two-fluid boundary layer,” J. Eng. Math. 199, 1–21 (2019).
I. V. Egorov, I. M. Ilukhin, and V. Ya. Neiland, “Numerical modeling of the interaction between shock wave and boundary layer past moving surface,” Fluid Dyn. 55, 110–117 (2020).
A. M. Gaifullin and A. V. Zubtsov, “Asymptotic structure of unsteady flow over a semi-infinite plate with a moving surface,” Fluid Dyn. 48, 77–88 (2013).
Te Ha Chzhun, S. I. Bezrodnykh, and V. B. Zametaev, “Incompressible boundary layer with counterflows at a given pressure gradient,” Comput. Math. Math. Phys. 62 (6), 249–255 (2022).
Vik. V. Sychev, “Laminar separation from a surface slowly moving upstream,” TsAGI Sci. J. 47 (3), 237–266 (2016).
M. A. Kravtsova, V. B. Zametaev, and A. I. Ruban, “An effective numerical method for solving viscous–inviscid interaction problems,” Philos. Trans. 363 (1830), 1157–1167 (2005).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bezrodnykh, S.I., Zametaev, V.B. & Chzhun, T.H. Singularity Formation in an Incompressible Boundary Layer on an Upstream Moving Wall under Given External Pressure. Comput. Math. and Math. Phys. 63, 2359–2371 (2023). https://doi.org/10.1134/S0965542523120060
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542523120060