Axially Symmetric Flow

Abstract

Perhaps the most basic, and consequently the most important, particular case of a spatial flow is an axially symmetric spatial flow, as for example flow past bodies of revolution. Let x, r, χ denote the cylindrical coordinates with the x-axis coincident with the axis of symmetry of the flow. The velocity vector then no longer depends on the rotation angle χ, but rather entirely on x and r, and always lies in a meridian plane (plane through the x-axis). As a result, the flow in all meridian planes is the same, and needs to be studied only in a given meridian plane. Below we shall apply the hodograph transformation. Thus, x, r denote the cylindrical coordinates in the physical plane in which the flow actually takes place, whereas the corresponding velocity components, u, v, determine the cylindrical coordinates in the hodograph plane. The velocity potential equation of an axially symmetric flow in the physical plane has the form: $$ \phi _{xx} (1 - \phi x^2 /a^2 ) + \phi _{rr} (1 - \phi r^2 /a^2 ) - 2\phi _{xr} \frac{{\phi _x \phi _r }} {{a^2 }} + \phi _r /r = 0, $$ (5.1.0) where φ is the velocity potential and the subscripts denote partial differentiation. As is known, the following relations hold: u = φ x , v = φ r . When transforming this equation into the hodograph plane, we introduce a new function Φ connected with φ by means of the formula: $$ \Phi = x\phi _x + r\phi _r - \phi . $$ (5.1.1)