6.1. 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 water depth changes smoothly along the length of the channel. A key factor here is the ratio of the change in depth (dy) to the change in length (dx), which is a measure of how steeply the depth varies. When this ratio is much less than one (dy/dx << 1), it indicates a slow and gentle change in depth over a long distance.

Think of a gently flowing river that meanders through a landscape. As you walk along the banks, you might notice that the water level rises and falls gradually, like soft hills. This gradual change in water depth over a long distance characterizes a gradually varied flow.

Assumptions behind gradually varied flow include: 1. The channel is prismatic, meaning the cross-sectional shape, size, and slope are constant. 2. The flow in the channel is steady and non-uniform. 3. The channel bed slope is small. 4. The pressure distribution at any section is hydrostatic. 5. Resistance to flow at any depth is given by the corresponding uniform flow equations, such as Manning's equation.

In analyzing gradually varied flow, there are several key assumptions to consider:

1. The channel needs to have a uniform cross-section (prismatic). This ensures that as the water flows, the characteristics of the channel don’t change abruptly.
2. The flow state is steady, meaning that over time, the flow rate remains consistent, but it is not uniform since depth varies.
3. The slope of the channel bed must be small to minimize changes in flow dynamics.
4. Hydrostatic pressure distribution is assumed, meaning that the pressure at any point in the fluid depends only on the height of the fluid above it.
5. Flow resistance can be calculated using uniform flow equations, where energy losses are consistent with what would occur in straight, uniform conditions.

Imagine a wide, slow-moving river with steady water levels. If the riverbed is like a channel that doesn’t change in shape or size, and the slopes are very gentle, we can predict how the water behaves as it travels downstream. This situation allows us to apply our understanding of gradually varied flow and make accurate calculations, similar to how engineers design stable bridges over a river.

When we differentiate the total energy equation with respect to x, we derive the energy slope, bottom slope, and the water surface slope. The resulting differential equation for gradually varied flow can be expressed as: dy/dx = (S0 - Sf) / (Q^2 T / g A^3).

To analyze the gradients and flows in a gradually varied flow situation, we look at the total energy in the water, which includes potential energy (height), kinetic energy (velocity), and pressure energy. By differentiating these energy components with respect to the length of the channel (x), we can determine how the energy changes over space. The resulting equation helps us calculate the change in depth (dy) as we move along the channel. The terms in this equation include:

• S0: bed slope,
• Sf: energy slope,
• Q: flow rate,
• T: top width of the channel,
• A: cross-sectional area of the flow. This equation assists in understanding how flow behaves under varying conditions.

Think of a water slide at a park. If the slide has a smooth incline without steep drops, as a child goes down, they gradually pick up speed while the height remains consistent. We can use similar calculations to understand how fast the water will flow at different points along the slide, which reflects the principles of gradually varied flow.

If the flow rate Q, Manning's number n, and S0 are fixed, then the normal depth y0 and the critical depth yc are also fixed. There are three relationships between normal depth y0 and critical depth yc: 1. y0 > yc (mild slope), 2. y0 < yc (steep slope), 3. y0 = yc (critical slope).

In the context of gradually varied flow, we can classify different profiles based on the relationship between normal depth (y0, the depth at uniform flow) and critical depth (yc, the depth at which flow changes from subcritical to supercritical). The classifications are:

1. Mild slope (y0 > yc): This is when the normal depth is greater than the critical depth, leading to stable subcritical flow.
2. Steep slope (y0 < yc): Here, normal depth is less than critical depth, causing supercritical flow which is unstable.
3. Critical slope (y0 = yc): At this point, flow transitions and is balanced between sub and supercritical states.

Picture a road on a hill. When the hill is not too steep (mild slope), cars move smoothly without much trouble. However, if the road is too steep (steep slope), cars might struggle to keep a steady pace or lose control if they go too fast. The critical slope is like the perfect incline where speeds are manageable without any abrupt changes.

Regions in the flow can be divided based on the relationships between normal depth and critical depth: Region 1 is above both CDL (critical depth line) and NDL (normal depth line), Region 2 is between CDL and NDL, and Region 3 is below NDL.

In gradually varied flow, flow regions help us visually understand how depth changes relative to critical and normal depths. These are divided as follows:

1. Region 1: Above both the critical depth line (CDL) and normal depth line (NDL). This area usually represents a stable subcritical flow zone.
2. Region 2: Located between the CDL and NDL, this area indicates transitional conditions where flow might shift from subcritical to supercritical.
3. Region 3: Below the NDL, where flow may be in a critical or unstable state, especially in adverse conditions.

Imagine a swimming pool with three sections: a shallow end (Region 3), a deep end (Region 1), and a middle section (Region 2). Depending on where you swim, the water depth and flow changes. Just as you navigate those depths, water flows through a channel with varying depth based on these regions.

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