On the flow in channels when rigid obstacles are placed in the stream

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Abstract
A previous paper drew attention to the collective importance of three physical quantities Q, R, S associated with ideal fluid flow in a horizontal channel. Invariability of these quantities at different cross-sections of the flow implies respectively conservation of flow rate, energy and momentum; and their values determine a wave-train uniquely. The properties of Q, R, S are recalled in the present paper to account for the various effects of lowering a rigid obstacle into a stream. The conditions giving rise to dissimilar types of flow are examined; in particular, the circumstances causing stationary waves on the downstream side are clearly distinguished from those under which the receding stream assumes a uniform ‘supercritical’ state. A well-known result in the theory of the solitary wave is shown to apply to the receding stream even when the extreme conditions for the wave are exceeded; although it fails to account for a region close to the obstacle where the curvature of the streamlines becomes large. In passing, a feature of the theory is shown to bear on the practical problem of producing a uniform stream. Precise calculations are made for the flow under a vertical sluice-gate and under an inclined plane. To account for the region near the bottom edge of the sluice-gate, a method based on conformal transformation is used whereby an unknown curve in the hodograph plane is approximated by an arc of an ellipse. The accuracy of the results is more than sufficient for practical purposes, and they compare favourably with solutions previously obtained by relaxation methods. A number of experiments with water streams are described.

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