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Okay class, today we will explore what exactly is meant by gradually varied flow. Can anyone summarize how we define it?
Is it when the flow depth changes gradually over a large length of the channel?
Exactly right! We consider this when $ rac{dy}{dx} $ is much less than 1. In simpler terms, the change in water depth is gentle across the length of the channel.
Now, let's talk about the key assumptions for analyzing gradually varied flow. Who can list some of these assumptions?
The channel needs to be prismatic, right?
Correct! Also, the flow should be steady and non-uniform. Can anyone explain what steady and non-uniform means?
Steady means that initially, the flow doesn’t change with time, but non-uniform means the depth changes with length?
Yes, great job! And we also need to assume small bed slopes and hydrostatic pressure distribution.
Next, we need to understand how we mathematically express gradually varied flow. Can anyone tell me the significance of total energy in this context?
Is it related to the height and velocity of the flow?
That's correct! Total energy combines both potential and kinetic energy. The formula we use is . What do these terms represent?
H is total energy, z is the elevation head, y is the depth of water, and V is the velocity.
Spot on! As we differentiate this equation, we find relationships among energy slope, channel slope, and flow depth.
Finally, let's classify flow profiles based on the relationships between normal depth and critical depth. What do we get when normal depth is greater than critical depth?
That’s a mild slope.
Correct! And what about when the normal depth is less than critical depth?
That’s a steep slope.
Absolutely! There is also a critical slope where they are equal. These classifications are vital for determining flow behavior.
Characterized by gradual changes in water depth over distance.
Includes channel prismatic shape, steady flow, small slope, and hydrostatic pressure distribution.
Categories based on depth relationships like mild slope, steep slope, critical slope, and conditions of horizontal or adverse slope.
In flows that barely change in line, Hydrostatic pressure makes it fine.
Imagine a gentle river, flowing down a calm slope, where the depth of water changes like a smooth blanket across its surface.
Remember the acronym P.H.S. - Prismatic shape, Hydrostatic pressure, Steady flow for assumptions.
F.S.C. - Flow Slope Classification for Flow profiles: Mild, Steep, Critical.
Term
What is gradually varied flow?
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Term
What assumptions are made in gradually varied flow?
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Flow regime in which the water depth changes gradually over a lengthy segment of the channel.
A channel shape where the cross-sectional geometry remains uniform along its length.
Slope of the energy line in a flow profile.
The depth of flow in a channel under uniform flow conditions.
The minimum depth of flow at which the specific energy is minimized.
A dimensionless number that compares inertial and gravitational forces in fluid flow.
Gradually varied flow refers to the non-uniform flow in open channels where the flow depth changes gradually over a long distance.
This section covers the definition, assumptions, and mathematical formulation of gradually varied flow. It introduces differential equations governing the flow and classifies flow profiles into mild slope, steep slope, and critical slope based on relationships between normal and critical depths.
In open channel hydraulics, gradually varied flow refers to situations where the flow depth changes gradually over a large length of the channel, indicated by $\frac{dy}{dx} $ being much less than 1. The significant assumptions underpinning this flow include a prismatic channel with a constant cross-sectional shape, steady and non-uniform flow conditions, a small channel bed slope, and a hydrostatic pressure distribution along with applying resistance equations from uniform flow, such as the Manning's equation.
The energy relationship for gradually varied flow is expressed in terms of total energy (H), which is derived from potential and kinetic energies. The mathematical representation is further developed into a differential equation that incorporates parameters such as top width (T), area (A), and flow rate (Q). Such equations help in determining the energy slope and water surface profile in varying channel depths.
Different categories of flow profiles are identified based on the relations between normal depth (y0) and critical depth (yc), leading to classifications of mild slope, steep slope, and critical slope, alongside situations such as horizontal bed and adverse slope. These characteristics play an essential role in predicting flow behavior in open channels.