6.3. Differential Equation of Gradually Varied Flow

The flow in a channel is termed as gradually varied, if the flow depth changes gradually over a large length of the channel. We said that, that dy by dx is very much less than 1.

Gradually varied flow refers to a situation in an open channel where the depth of the water changes slowly along the length of the channel. The rate of change of depth (indicated by dy/dx) is small, suggesting that the flow does not undergo abrupt changes but transitions smoothly over a significant distance.

Imagine a gently sloping river where the water level gradually rises from one end to the other rather than a waterfall where the water abruptly drops. The gradual slope of the river represents gradually varied flow where the change in water depth is smooth over a distance.

The first assumption is that the channel is prismatic, meaning the cross-sectional shape, size, and the bed slope are constant. Second assumption is that the flow in the channel is steady and non-uniform. The third one is the channel bed slope is small.

When analyzing gradually varied flow, several assumptions are made: 1. The channel must have a prismatic shape, meaning it has a constant cross-section throughout, like a straight pipe. 2. The flow should be steady, meaning that the depth at any point does not change over time, though it may change in space. 3. The slope of the channel bed is small, indicating that the channel does not have steep inclines, which simplifies the analysis.

Think of a water slide that is perfectly straight and smooth. The channel's prismatic shape allows water to flow steadily down without unexpected changes in depth or direction, similar to how gradually varied flow operates in a simple, consistent channel.

The total energy H of a gradually varied flow can be expressed as: H = z + y + α V²/2g. If we assume α = 1, we get H = z + y + V²/2g.

In gradually varied flow, the total energy of the water (H) at any point is the sum of the potential energy (z), the kinetic energy component (V²/2g), and the depth of the water (y). By assuming α equals 1, we simplify the equation to a scenario where these energies are directly additive. This equation helps in understanding how energy varies through the flow.

If you think of a water tank, the height of the water (potential energy), the flow speed (kinetic energy), and the overall depth of water provide a total energy amount. This total energy changes based on how much water you have and how fast it's flowing, very much like how the flow behaves in a channel.

The three terms dH/dx, dz/dx, and dy/dx represent the energy slope, bottom slope, and water surface slope respectively. They are critical to understand the flow behavior.

In the derivation of the differential equation for gradually varied flow, we encounter three critical slopes: 1. The energy slope (dH/dx) provides insight into how the total energy changes along the flow. 2. The bottom slope (dz/dx) reflects changes in the channel bed elevation. 3. The water surface slope (dy/dx) indicates how the water surface level varies against the bed. Understanding these terms is essential to characterize the flow in channels.

Consider a gentle stream. The slope of the bed (the ground underneath), the slope of the water level, and the overall energy of the water moving down the stream gives a comprehensive viewpoint on how the water is flowing. Observing all three slopes helps predict how quickly the water will move and whether it will overflow.

Based on fixed parameters, there can be three relationships between normal depth y0 and critical depth yc: y0 > yc, y0 < yc, and y0 = yc.

The relationship between normal depth (y0) and critical depth (yc) helps classify flow profiles into three categories: 1. If y0 is greater than yc, the flow is termed subcritical. 2. If y0 is less than yc, the flow is supercritical. 3. If they are equal, critical flow is happening. These classifications are crucial when designing channels.

Imagine a highway. If traffic (flow) is moving at a slower pace than the speed limit, that's like subcritical flow. If traffic is rushing by at high speeds, that's similar to supercritical flow. When traffic is flowing exactly at the speed limit without any interruption, that's akin to critical flow.

The channels can be classified into mild slope (y0 > yc), steep slope (y0 < yc), critical slope (y0 = yc), horizontal bed (S0 = 0), and adverse slope (S0 < 0).

Channels can exhibit five classifications based on their slopes and depth relations: 1. Mild slope channels have normal depths exceeding critical depths. 2. Steep slope channels have normal depths lesser than critical depths indicating supercritical flow. 3. Critical slope channels have normal depth equal to critical depth. 4. Horizontal bed channels have no normal depth due to being level. 5. Adverse slope channels see a downward slope leading to no normal depth. Knowing these types helps engineers design effective channels.

Picture different types of roads and their functionalities. A mild slope is like a gentle hill allowing easy driving; steep slopes might be like fast steep highways; a horizontal bed is like a flat boulevard; and adverse slopes could represent roads that decline unexpectedly, causing potential hazards for drivers.

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