Understanding Ogee Spillway Discharge Proportionality

An ogee spillway is a type of overflow spillway often used in dams. Its crest is shaped like an 'S' or ogee curve, designed to follow the natural trajectory of the overflowing water. This shape helps to minimize negative pressure on the crest and maintain smooth flow, which is important for efficiency and preventing cavitation.

The flow over the crest of an ogee spillway, when operating under design head, is similar to the flow over a broad-crested weir or, more closely, a sharp-crested weir. The fundamental principle governing discharge over a weir or spillway crest relates the flow rate to the head of water above the crest.

Discharge Formula for Weirs and Spillways

The general formula for the discharge rate (Q) over a weir or spillway is given by:

$$Q = C \cdot L \cdot H^n$$

Where:

• $Q$ is the total discharge rate (volume per unit time).
• $C$ is a discharge coefficient, which accounts for various factors like the shape of the crest, approach velocity, and viscosity.
• $L$ is the effective length of the spillway crest across which flow occurs.
• $H$ is the total head of water over the spillway crest (usually including approach velocity head, though often approximated by the static head above the apex for simplicity in proportionality questions).
• $n$ is an exponent that depends on the type of weir or spillway.

Proportionality with Head (H)

For a typical weir like a rectangular weir, the theoretical value of the exponent $n$ is derived to be 1.5. This value arises from integrating the velocity profile over the depth of flow, where the velocity is proportional to the square root of the head ($v \propto \sqrt{y}$) and integrating this over the depth ($y$) from 0 to $H$.

For an ogee spillway operating under free flow conditions, the flow characteristics over the crest resemble those of a weir. Therefore, the discharge formula for an ogee spillway is commonly expressed in a similar form:

$$Q = C_d \cdot L \cdot H_e^{1.5}$$

Where:

• $Q$ is the discharge.
• $C_d$ is the discharge coefficient for the ogee spillway.
• $L$ is the effective length of the spillway crest.
• $H_e$ is the effective head over the crest (often the total head, including approach velocity head).

The question asks for the discharge passing over an ogee spillway *per unit length* of its apex line, which means we are looking at $Q/L$. Let $q = Q/L$ be the discharge per unit length.

From the formula, the discharge per unit length is:

$$q = C_d \cdot H_e^{1.5}$$

Assuming $H$ in the question represents the effective head $H_e$ over the apex of the crest, and since $C_d$ is a coefficient (treated as constant for proportionality), the discharge per unit length ($q$) is directly proportional to $H^{1.5}$.

Thus, the discharge passing over an ogee spillway, per unit length of its apex line, is proportional to $H^{1.5}$.

Revision Table: Key Concepts

Additional Information: Factors Affecting Ogee Spillway Discharge

While the proportionality $q \propto H^{1.5}$ is fundamental, the actual discharge coefficient $C_d$ for an ogee spillway is not strictly constant. It can vary depending on several factors, including:

• The ratio of actual head to the design head.
• The slope of the upstream face of the dam.
• The height of the spillway crest above the upstream bed.
• Submergence from the downstream tailwater level.
• The formation of negative pressure on the crest at heads greater than the design head.

Engineers use design charts and models based on experimental data (like those from the U.S. Army Corps of Engineers) to determine appropriate $C_d$ values for specific ogee spillway designs and operating conditions, but the basic proportionality with $H^{1.5}$ remains the core relationship for free flow.