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Three-dimensional laminar boundary layer on a blunt body

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Abstract

We consider the Prandtl laminar boundary layer which occurs with stationary flow about a blunted cone at an angle of attack. The solution of the Prandtl equations is sought using a finite difference method. It is found that a smooth solution of the problem exists only in the vicinity of the rounded nose of the body, while far from the nose the solutions acquire a singularity; in the problem symmetry plane (on the downwind side) there is a discontinuity of the first derivatives of the velocity components and the density.

In the study of the Prandtl boundary layer in the problem of stationary flow about a pointed cone at an angle of attack, it has been shown [1] that the self-similar solution (dependent on two independent variables) of the Prandtl equations has a discontinuity of the first derivatives in the problem symmetry plane (on the downwind side of the cone). The suggestion has been made that in the three-dimensional problem of flow about a blunt cone at an angle of attack the solutions of the Prandtl equations may also be discontinuous. The present study was carried out to clarify the nature of the behavior of the solutions of the three-dimensional Prandtl equations. To this end we considered stationary supersonic flow of an ideal gas past a blunted cone. The results of this study (as well as those of [1]) were obtained using a numerical, finite-difference method. However, an analysis of the numerical results (investigation of the scheme stability, reduction of step size, etc.) shows that the properties of the solutions of the finite-difference equations are not in this case a result of numerical effects, but reflect the behavior of the solutions of the differential equations.

The mathematical problem on the boundary layer which is considered in this study will be formulated in §2; this formulation is due to K. N. Babenko.

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Abbreviations

x,θ, z:

coordinates fixed to the body surface (x is directed along the body generator

z:

perpendicular to the surface

θ :

angular coordinate in the plane perpendicular to the body axis

x:

measured from the vertex, z from the surface, from the upward body symmetry half-plane)

l :

distance from the plane x=const to the body vertex

u, v, w:

velocity components in the x,θ, z directions, respectively

ρ :

density

T:

temperature

p:

pressure

η, ϰ:

viscosity and thermal conductivity coefficients

cp :

specific heat

References

  1. N. D. Vvedenskaya, “Calculation of the boundary layer on a cone at an angel of attack”, Zh-1 vych. matem. i matem. fiziki, vol. 6, no. 2, 1966.

  2. K. I. Babenko and V. V. Rusanov, “Difference methods of solving three-dimensional gasdynamic problems”, collection: Proceedings of All-Union Congress on Theoretical and Applied Mechanics (survey papers) [in Russian], Izd-vo Nauka, 1965.

  3. A. L. Ankudinov, “Results of the calculation of the boundary layer on blunt cones in a supersonic flow”, Zh-1 vych. matem. i matem. fiziki, vol. 5, no. 5, 1965.

  4. V. V. Paskonov, “Standard program for solving boundary layer equations”, collection: Numerical Methods in Gasdynamics [in Russian] (collection of works of the Computer Center of Moscow University), Izd-vo MGU, no. 2, 1963.

  5. V. M. Paskonov and N. V. Rabinkina, “Solution of the equations of the unsteady boundary layer by a difference method”, collection: Numerical Methods in Gasdynamics [in Russian] (collection of works of the Computer Center of Moscow University), Izd-vo MGU, no. 4, 1965.

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Authors and Affiliations

  1. Moscow

    N. D. Vvedenskaya

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  1. N. D. Vvedenskaya
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Additional information

The author wishes to thank K. I. Babenko and E. E. Shnol for detailed discussions, M. G. Orlov of the Mathematics Institute for compiling the computer program, and L. N. Evgrafov of the institute for assistance in preparing the manuscript.

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Vvedenskaya, N.D. Three-dimensional laminar boundary layer on a blunt body. Fluid Dyn 1, 25–28 (1966). https://doi.org/10.1007/BF01022145

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  • DOI: https://doi.org/10.1007/BF01022145

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