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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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