6.2. Assumptions behind 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 flow depth changes slowly and uniformly across a considerable length of the channel. This is mathematically expressed by saying that the slope of the depth change (denoted as dy/dx) is much less than 1. In simpler terms, if you were to look at the channel, the depth would not fluctuate abruptly; instead, it would change gradually, like a gently sloping hill, rather than a steep cliff.

Imagine a long, winding river where, as you walk along its banks, the water level rises or falls gently. This gradual change in water level resembles the slowly rolling hills of a landscape, contrasting sharply with a steep waterfall that could represent rapidly varied flow.

The first assumption is that the channel is prismatic, which means the cross-sectional shape, size, and the bed slope are constant.

When we say the channel is prismatic, we mean that the channel's shape and slope do not change along its length. This homogeneity simplifies calculations and helps in predicting the flow characteristics effectively. For example, if the channel has a rectangular cross-section with a consistent width and depth, the fluid dynamics can be analyzed more straightforwardly without concern for varying shapes or sizes.

Think of a water slide in an amusement park. If the slide maintains the same width and slope all the way down, it's easier to predict how fast the water will flow. In contrast, if the slide suddenly changed shape, like morphing from wide to narrow, predicting the speed and flow becomes complicated.

The second assumption is that the flow in the channel is steady and non-uniform. Non-uniform means that dy by dx is not equal to 0, while steady means dy by dt is 0.

In the context of flow, 'steady' indicates that at any given point in the channel over time, the depth of the water and the flow velocity remain constant. However, 'non-uniform' means that this depth does not remain the same across the channel's length – it varies gradually. Thus, while you might see a steady flow of water at different depths in various sections of the channel, the water levels themselves are changing as you move along the channel.

Picture a long, inflation tube that you fill with air. At the start of the tube, the air pressure is consistent and stable, but as the tube stretches out, the pressure gradually decreases as you move further along. The air flow is steady, but the pressure (reflecting how high the water might be in our channel situation) varies over the length of the tube.

The third assumption is that the channel bed slope is small.

This assumption implies that the angle of the channel bed is gentle. If the bed slope were too steep, the flow dynamics would change significantly, affecting flow depth and velocity. A small slope allows for the gradual variation of flow depth as the water moves downstream, which is a key characteristic of gradually varied flow.

Consider a long, gentle ramp leading into a pool. If the ramp is steep, water spills in quickly and unevenly, creating splashes and turbulence. However, with a gentle slope, the water flows smoothly and gradually into the pool, which allows for stable and predictable water levels.

The approximation assumption is that the pressure distribution at any section is hydrostatic.

This assumption states that the pressure in the fluid at rest is distributed according to the height of the fluid column above the point being measured. In simpler terms, at any given depth in the channel, the pressure exerted by the water above is uniform and increases with depth, akin to the way atmospheric pressure increases with altitude in the atmosphere.

Imagine a tall glass filled with water. The water pressure at the bottom of the glass is greater than that at the top due to the weight of the water above it. Similarly, in a stable flow of river water, the pressure at any point below the surface will be influenced by the weight of the water above it, leading to a predictable pressure increase with depth.

The resistance to the flow at any depth is given by the corresponding uniform flow equation.

In gradually varied flow, it is assumed that the resistance to flow can be approximated using uniform flow equations like Manning's or Chezy's equation. These equations help estimate the resistance based on various factors like channel geometry and flow conditions, assuming they hold true even when the flow is gradually changing.

Think of riding a bicycle on a flat terrain versus on a bumpy surface. On a flat surface, you can apply uniform effort to maintain speed, similar to how resistance is constant in uniform flow conditions. On a bumpy path, however, the effort needed to maintain the same speed varies, like how flow resistance can change in gradually varied flow, but we can still use our knowledge of flat surfaces to estimate our performance.

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