2.3. Problem 2: Gradually varied flow profile

A rectangular channel with a bottom width of 4 meter and a bottom slope of 0.0008 has a discharge of 1.5 meters cube per second. In a gradually varied flow in this channel, the depth at certain location is found to be 0.30 meter. Assuming the Manning's n is equal to 0.016, determine the type of gradually varied profile.

In this problem, we are dealing with a rectangular channel that has specific dimensions, a defined slope, and a certain discharge rate. We need to analyze the flow conditions to identify the type of gradually varied flow profile present. The known parameters include the bottom width of the channel (4 meters), bottom slope (0.0008), discharge (1.5 cubic meters per second), and Manning's roughness coefficient (0.016). The depth of water at a specific location in the channel is also provided (0.30 meters). Our goal is to find out if this flow is characterized as mild slope, steep slope, critical slope, or horizontal bed.

Imagine a water slide at a theme park that starts off steep and then gently flattens out. The way water flows down that slide changes depending on how steep or flat the slide is—just like the water flow in this problem changes with the slope of the channel. Understanding the channel's parameters helps us determine how the water moves through it.

In the normal, I mean, you remember, the way we calculated the critical depth yc was q square by g to the power 1 by 3, if you remember the formula from our lectures last week. So, this is going to be 0.0375 whole square by, g is 9.81, and whole to the power 1 by 3. If you do this calculation using your calculator or, you know, by hand, this is going to be 0.243 meter.

To find the critical depth (yc) in the channel, we can use the formula that relates discharge per unit width (small q), gravity (g), and critical depth. Given that the small q is 0.0375 cubic meters per second per meter, we can square this value and divide by the gravity constant (9.81 m/s²) raised to the power of one-third. Completing this calculation yields a critical depth of approximately 0.243 meters.

Think of the critical depth like the turning point of a roller coaster. Just like the height at which the cart begins to pick up speed and create thrilling drops, critical depth determines how water behaves in the channel. If the depth at which the water is flowing is less than the critical depth, it might flow more rapidly or be less turbulent, akin to a roller coaster that hasn't reached its peak yet.

Now, we can use the Manning’s equation and see, 1 by n A into R to the power 2 by 3 into S0 to the power 1 / 2. So, we see, area we already know, so if in this is a rectangular channel, so this is the y and this is b, so area is b y0 already, because if the, if we also need to calculate the normal depth, so area is going to be by0.

Manning's equation allows us to determine the frictional flow in the channel. In it, 'n' is the Manning's roughness coefficient (0.016), 'A' stands for the cross-sectional area (width multiplied by the depth), and 'R' is the hydraulic radius. For a rectangular channel, the area can be calculated simply as the product of the bottom width (b) and the depth (y0). This equation helps us analyze how the water will flow based on how 'rough' the channel is due to its surface texture.

Consider how fast water flows in different terrains: water flows quickly down a smooth, slick surface, like a polished slide, and slows down on a rough, rocky pathway. Manning's equation helps predict how quickly water flows in our channel depending on its 'roughness' and flow conditions.

Since normal depth is greater than yc, this is going to be an M slope, first of all, M slope channel. Now, the water depth is actually 0 point, so actually water depth is not 0.03, which I wrote, it was 0.3 meter. So, we see this, the y0 is greatest, greater than y and so this is critical depth, this was 0.426, this was 0.3 and this was, yc was 0.243.

Now that we have our normal depth (y0) and critical depth (yc), we can classify the type of flow profile. In this scenario, the normal depth (0.426 meters) is greater than the critical depth (0.243 meters). This indicates that the channel is categorized as mild slope. Additionally, since the actual depth of water (0.30 meters) lies between normal depth and critical depth, it reinforces that this is an M2 curve type of flow profile.

Think of this step as creating a mountain range map with different elevations—the critical depth represents valleys, while the normal depths represent hilltops. Knowing how these heights relate helps us determine whether the land slopes upward (mild slope) or downward, which directs water flow just as water will choose easier paths on a landscape.

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