Solid boundaries and image systems

The fact that the flow is always along a streamline and not through it has an important fundamental consequence. This is that a streamline of an inviscid flow can be replaced by a solid boundary of the same shape without affecting the remainder of the flow pattern. If, as often is the case, a streamline forms a closed curve that separates the flow pattern into two separate streams, one inside and one outside, then a solid body can replace the closed curve and the flow made outside without altering the shape of the flow (Fig. 3.12a). To represent the flow in the region of a contour or body it is only necessary to replace the contour by a similarly shaped streamline. The following sections contain examples of simple flows which provide continuous streamlines in the shapes of circles and aerofoils, and these emerge as consequences of the flow combinations chosen.

When arbitrary contours and their adjacent flows have to be replaced by identical flows containing similarly shaped streamlines, image systems have to be placed within the contour that are the reflections of the external flow system in the solid streamline.

Figure 3.12b shows the simple case of a source A placed a short distance from an infinite plane wall. The effect of the solid boundary on the flow from the source is exactly represented by considering the effect of the image source A! reflected in the wall. The source pair has a long straight streamline, i. e. the vertical axis of symmetry, that separates the flows from the two sources and that may be replaced by a solid boundary without affecting the flow.

Figure 3.12c shows the flow in the cross-section of a vortex lying parallel to the axis of a circular duct. The circular duct wall can be replaced by the corresponding streamline in the vortex-pair system given by the original vortex В and its image B’.

distance of the vortex axis from the centre of the duct.

More complicated contours require more complicated image systems and these are left until discussion of the cases in which they arise. It will be seen that Fig. 3.12a, which is the flow of Section 3.3.7, has an internal image system, the source being the image of a source at — oc and the sink being the image of a sink at +oc. This external source and sink combination produces the undisturbed uniform stream as has been noted above.