2.2. Horizontal Bottom Assumption

Let us say the bottom is described as z = - h of x. So, if there is you know this is the riverbed or the seabed and this can be z here. So, if this is x and this is z. So, this depth z = - h f x because, we are considering the 0 at the free surface if we consider 0 at the free surface. Since from the boundary condition we see the bottom is fixed for you dot n is going to be 0.

This chunk introduces the bottom boundary condition in hydraulic engineering. The bottom is defined mathematically as z = -h(x), which indicates the depth below the surface. In this context, the origin is at the still water level. Based on boundary conditions from prior studies, if the bottom is fixed, the normal velocity at the boundary, represented as u·n, will be zero.

Imagine the ocean floor as the bottom of a swimming pool. The water surface is the 'free surface' above it, and the depth from this surface to the bottom can be thought of as 'h'. If we say that the pool has a rigid bottom, then you cannot have any water moving through the floor, similar to how we assume velocities will be zero at the seabed.

Now, if we assume a case of horizontal bottom, we know that the height will not vary as a function of x. Therefore, dh/dx is going to be 0 for horizontal bottom, which we have written here. Therefore this means w = 0.

Here, the text delineates a critical assumption in hydraulic studies regarding horizontal bottoms. In this case, the depth does not change; therefore, the gradient of height with respect to x (dh/dx) equals zero. This leads to the conclusion that the vertical velocity component (w) downwards at this bottom boundary condition must also be zero. This simplification aids in analyzing fluid behavior.

Think of a calm lake where the water surface is perfectly level with no slopes or waves. If you were to take a measurement of how 'deep' the lake is at various points along the shore, you'd find that, assuming the shape of the bed is flat, the depth remains constant – just like our horizontal bottom assumption.

So, we have now proved that the boundary w = 0 using the bottom boundary condition and we have derived everything from the basics of the boundary condition. If we have a sloping bottom, we can simply in case of a sloping bottom like this, we can write w by u = - dh dx very simple.

Establishing that w = 0 is essential for fluid dynamics because it allows us to analyze flows where the bottom is horizontal without complicating the equations. Upon changing to a sloping bottom, the relationship between w and u is captured in the relation w/u = -dh/dx, which means that any change in height directly influences vertical velocity—a key concept in understanding fluid motion over surfaces of varying elevation.

Consider water flowing down a gentle hill versus a flat surface. On the flat surface, there is no vertical movement of water (w = 0); on the hill, the steeper the slope (dh/dx), the faster water flows downward. This illustrates how the slope impacts movement, much like how slopes and levels impact fluid mechanics in our studies.

The question to you is, could we treat this bottom as a streamline or not? Yes, since the flow is everywhere tangential to it.

The concept of treating the bottom as a streamline implies that the flow direction is aligned with the surface of the bottom boundary. Since we established that w = 0, the flow acts along the surface, validating this treatment. Streamlines are essential in fluid dynamics as they help simplify complex flow patterns by illustrating the direction of fluid motion.

Imagine a river flowing quietly over a flat riverbed. The water moves parallel to the riverbed, indicating that the riverbed is acting as a streamline. Thus, both its edge and bottom define how the water flows, just as in our model, where we equate the bottom condition to a streamline because of the tangential flow.

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