5. Lecture – 34

Welcome students. This is the 7th lecture for this broad topic, that is, open channel flow and as we were going to start this gradually varied flow as promised in the last lecture. Until now, we have studied that in open channel flow with the classification based on space, the dimensions, I mean, there are 3 types of flows; one is uniform flow and the other is non-uniform flow. So, non-uniform have 2 different categories; the first is gradually varied flow and the second is rapidly varied flow.

In this introductory chunk, the instructor sets the stage for discussing gradually varied flow within the context of open channel flow. Here, he distinguishes between three types of flow: uniform flow, non-uniform flow, and then highlights that non-uniform flow can further be categorized into gradually varied flow and rapidly varied flow. This establishes a foundational understanding necessary for delving deeper into the topic.

Think of a river flowing downstream. When the water flows at a consistent depth and speed, that's uniform flow. However, when it encounters rocks, bends, or changes in elevation, the water's depth and speed change gradually; this changing flow is similar to how our emotions change gradually as events unfold in our lives.

So, to get started, we should understand what exactly a gradually varied flow is. We have already derived an equation before, but to make it more clear we will see it in a little different way, a derivation of a different sort. So, what is 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, dy by dx is very much less than 1.

Gradually varied flow is defined by the gradual change in flow depth over an extended section of the channel. This can be quantitatively expressed with the derivative 'dy/dx', which indicates how quickly the flow depth changes with respect to distance along the channel. A value of 'dy/dx' that is much less than 1 signifies that the change in depth is gentle or gradual, reflecting steady flow conditions.

Imagine sliding down a gentle hill versus a steep cliff. The gentle slope allows you to descend slowly, representing gradually varied flow, while the steep cliff would result in a quick drop, similar to rapid flow where changes are abrupt and not gradual.

So, what are the assumptions behind gradually varied flow? First, that the channel is prismatic, this means 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. Non-uniform means that dy by dx, steady means, dy by dt is 0 but dy by dx is not equal to 0. The third assumption is the channel bed slope is small.

There are several assumptions necessary for analyzing gradually varied flow: 1) The channel must be prismatic, meaning the shape and size don't change, allowing for consistent behavior; 2) The flow should be steady, implying that changes in flow depth don't occur over time (dy/dt = 0), but they can vary along the length (dy/dx not equal to 0); 3) Bed slope should be small to ensure that flow changes are gradual rather than sudden.

Think about a bathtub filled with water. If the shape of the tub is consistent and you gently tilt it, the water level will gradually change. Similarly, if the bathtub's sides were steep (like a canyon river), the water depth would suddenly drop, analogous to rapidly varied flow. Keeping the slope gentle allows for gradual changes.

Apart from that, the resistance to the flow at any depth is given by the corresponding uniform flow equation. For example, uniform flow equations like the Manning's equation. You have to remember that in uniform flow equations energy slope Sf is used in place of bed slope S0.

A critical assumption for gradually varied flow is that we apply uniform flow resistance equations to it. For example, Manning's equation, which is a standard formula used to calculate flow rates in open channels. Notably, the energy slope (Sf) replaces the bed slope (S0) in these applications when analyzing flow at different depths.

Think of wind blowing through trees in an open field versus in a dense forest. In the field, the wind's speed represents uniform flow (steady and constant), while in the forest, the effects of branches and trunks additionally mirror how resistance works in different flow scenarios—despite varying conditions, we can still understand flow dynamics using foundational formulas.

Now, what the going to the differential equation of the gradually varied flow. So, first we have to draw, you know, figure the; I have shown you a figure here, where there is a small bed slope, there is a channel with a small bed slope S 0...

To analyze gradually varied flow mathematically, we derive a differential equation based on energy principles. This involves understanding relationships between energy slope, bed slope, and water surface slope. By drawing a diagram, the instructor can illustrate how these slopes interact, ultimately leading to a derived formula that describes the relationship between these factors and the flow behavior.

When driving along a curvy mountain road, the changes in slope of the road (akin to bed slope, energy slope, and surface slope) affect how fast or slow your vehicle travels. If the slope is steep, you may need to slow down; similarly, water behaves according to the slopes in a channel—understanding these relationships allows for predictions about its flow.

So, now the classification of flow profiles. What are the flow profiles in gradually varied flow? So, if the flow rate Q, Manning's number n and S0 are fixed...

Flow profiles in gradually varied flow can be classified depending on the relationships between normal depth (y0) and critical depth (yc). The instructor explains that these relationships can yield three distinct conditions: normal depth greater than critical depth (subcritical flow), less than (supercritical flow), or equal (critical flow). This classification has implications for predicting water behavior in channels.

Consider a water slide: if the angle of the slide is too shallow (greater than critical depth), the ride is gentle and slow (subcritical flow). If it’s too steep (less than critical depth), you’ll zip down fast (supercritical flow). Understanding these profiles helps in designing channels for desired flow velocities.

So, based on these, the channels can be classified into 5 categories...

Channels can be classified into five distinct categories based on the relationships between normal and critical depth, including mild slope (M), steep slope (S), critical slope (C), horizontal bed (H), and adverse slope (A). Each category influences flow behavior differently, such as maintaining uniform flow or not, which is crucial for hydraulic engineering applications.

Imagine different playground slides: a gentle slide (mild slope) that kids can easily use, a steep slide (steep slope) that gives a thrilling ride, and no slide at all (horizontal or adverse slope) which may only make it hard to slide down. This familiarity can help students visualize how channel conditions affect water flow.

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