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Unsteady disturbances of streaming motions around bodies

Published online by Cambridge University Press:  26 April 2006

H. M. Atassi  and
J. Grzedzinski
H. M. Atassi
Affiliation:
Department of Aerospace and Mechanical Engineering and Center for Applied Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
J. Grzedzinski
Affiliation:
Department of Aerospace and Mechanical Engineering and Center for Applied Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
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Abstract

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.

For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.

For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 385 - 403
Copyright
© 1989 Cambridge University Press

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References

Atassi, H. M. 1984 The Sears problem for a lifting airfoil revisited - new results. J. Fluid Mech. 141, 109–122.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 1, 83–103.Google Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433–468.Google Scholar
Goldstein, M. E. & Atassi, H. 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J. Fluid Mech. 74, 741–765.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech. 61, 625–706.Google Scholar
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1, 31–53.Google Scholar
Prandtl, L. 1933 Attaining a steady air stream in wind tunnels. NACA Tech. Memo. no. 726.Google Scholar
Ribner, H. S. & Tucker, M. 1953 Spectrum of turbulence in a contracting stream. NACA Rep. no. 1113.Google Scholar
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical applications. J. Aero Sci. 81 (3), 104–108.Google Scholar
Taylor, G. I. 1935 Turbulence in a contracting stream. Z. angew. Math. Mech. 15, 91.Google Scholar
Tikhonov, A. N. & Samarskii, A. A. 1963 Equations of Mathematical Physics. Macmillan.