Rayleigh Wave Theory: Gate Excitation Mechanism

Introduction

For hydraulic gates with free discharge and limited streamwise extent of the gate lip, viscous effects are usually negligible. In such cases, the dynamic forces acting on the gate are substantially due to pressure fluctuations in the upstream reservoir. In this chapter, the mechanism for generating fluctuating reservoir pressures will be explored.

Vibrating structures piercing the free surface of a fluid can induce surface waves that travel outward from the structure and carry energy away from the structure. The energy input into these waves is dissipated, or dispersed, as the waves travel away from the structure. These waves are called dispersive waves. In this chapter, the mathematical description of these dispersive waves will be developed.

Dispersive water waves emanating from a vibrating gate are shown in Figures 1 and 2. The Tainter gate, shown in Figure 1, is suspended on wire cables, which are elastic and play the role of a spring, permitting vibration of the whole gate as a rigid body around the trunnion pin center. The skinplate is sufficiently large and thin to undergo streamwise bending vibration with nodes on each spanwise side.

The long-span gate, shown in Figure 2, can also undergo streamwise bending vibration due to its limited streamwise thickness and its elasticity. It can also vibrate in the normal direction (up-and-down vibration), due both to the gate elasticity and to the spring effect of its wire suspension cables.

The whole gate vibration around the trunnion pin of the Tainter gate and, similarly, the up-and-down vibration of the long-span gate each cause a variation in the flow rate (also variously called ‘fluctuating discharge,’ ‘discharge fluctuation,’ or ‘discharge variation’) beneath the gate. When the long-span gate is inclined toward the downstream direction, or when the angle between the tangent to the Tainter gate skinplate and the gate crest tangent is less than 90°, the streamwise bending vibration of the gate also results in flow rate variation beneath the gate. Depending on the inclination of the skinplate or long-span gate, streamwise gate bending can result in a larger gate opening in the center of the channel, resulting in a discharge that varies with the streamwise motion of the gate.

Figure 1.

Dispersive water waves can be generated by both normal (up-and-down) and streamwise vibrations of a Tainter gate on a dam crest with flow from right to left

Figure 2.

Dispersive water waves generated by normal (up-and-down) and streamwise vibrations of a long-span gate with flow from right to left

The flow rate variation beneath the gate produces a fluctuation in the reservoir depth, and induces upstream propagating dispersive waves on the free surface. A similar argument can be made for streamwise gate vibrations, which may produce discharge fluctuations beneath the gate depending on the gate orientation relative to the crest. However, in addition, streamwise vibration will always produce a periodic fluctuation of the reservoir depth, due to the push-and-draw effect of the gate against the backwater, thereby inducing dispersive water waves on the free surface.

For thin long-span plates and thin Tainter gate skinplates, the reservoir flow field is independent of shed vortices; other viscous effects are sufficiently small to neglect. The fluctuating reservoir free surface due to small amplitude vibration of a gate can be considered, to a very good approximation, an inviscid flow to be analyzed with potential flow theory. However, in order to predict the dispersive nature of the water waves resulting from the damping effect of the fluid viscosity, the potential theory needs a certain modification to represent the damping effect.