5.1. Conditions for Flow in X and Y Directions

Let us say the bottom is described as z = -h of x. So, if there is this is the riverbed or the seabed and this can be z here. If we consider 0 at the free surface, then here origin is located at still water level, that is the surface of the water.

In this chunk, we learn about the bottom boundary condition in hydraulic engineering, where the seabed is represented by the equation z = -h(x). This means that the seabed's depth h varies with the horizontal position x. The current convention places the zero reference point at the free surface of the water, which simplifies our calculations by clearly defining a baseline (still water level) against which depth can be measured.

Think of a swimming pool where the surface of the water is the still water level. When someone dives into the pool, the maximum depth of the pool will change depending on the design (shaped bottom); similarly, here the seabed's depth (h) changes based on the horizontal position (x), illustrating how water bodies like rivers or lakes can have uneven bottoms.

Since from the boundary condition, we see the bottom is fixed for u dot n = 0. Therefore, if we assume u is a combination of small ui + w k, we can analyze the velocities in both x (u) and z (w) directions.

This section discusses how we apply boundary conditions to our fluid flow equations. Specifically, we establish that at the bottom boundary, the velocity component normal to the seabed (u dot n) must be zero, meaning there is no vertical velocity when water meets the fixed seabed. When we express the velocity u as a combination of two components (ui for the x-direction and wk for the z-direction), it allows us to analyze fluid motion in both horizontal and vertical contexts.

Imagine standing at the edge of a river; the water flows horizontally, but where the water meets the ground, there’s no upward or downward movement of the water at that point. It’s the same concept here; the velocity of water at the seabed doesn't push upwards.

In case of a sloping bottom, we can write w by u = -dh/dx.

Here, we explore how the conditions change when the seabed is not flat, but instead has a slope. This relationship indicates that the vertical velocity (w) relative to the horizontal velocity (u) is linked to the slope of the seabed (dh/dx). The negative sign implies that as the bottom of the water body slopes down (increases dh/dx), the vertical flow decreases, showing a direct relationship between the seabed slope and vertical fluid movement.

Picture a waterslide at a water park. As you slide down, gravity pulls you down quicker due to the slope. Similarly, when water flows over a sloped seabed, the angle of the slope affects how fast the water can move down, which is what this equation represents.

There is something called dynamic free surface boundary condition, meaning that the free surface of water can distort and isn't fixed.

The dynamic free surface boundary condition is critical in fluid dynamics because it describes how the surface of the water can change due to external forces or disturbances (like waves). Unlike a fixed bottom that holds water at a certain level, the surface can rise and fall, reflecting changes in pressure and flow dynamics. This flexibility must be modeled accurately to predict water movement properly.

Consider waves at the beach; the water surface constantly changes as waves come and go, moving up and down with the energy of the sea. This dynamic behavior is what engineers must account for when analyzing water flow conditions.

Kinematic free surface condition can be written down as u dot n = - (∂η/∂x · u + ∂η/∂y · v) + w.

This represents the relationship between the flow velocities and the surface elevation changes using derivatives. Essentially, it provides a mathematical way to relate how the water moves over time concerning its surface shape, where η stands for surface elevation. As the water flows, both its horizontal (u, v) and vertical (w) components influence the surface’s shape.

Think of a duck floating on the surface of a pond. As it swims, the ripples and waves in the water change (η), and these movements can be calculated by considering how fast the duck moves through the water (u, v) and how the water level fluctuates (w) due to its weight and the surface ripples.

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