2. Bottom Boundary Conditions (BBC)

So, 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. So, here origin is located at still water level that is the surface of the water.

The bottom boundary conditions (BBC) refer to the conditions that define the behavior of fluid at the bottom of a water body, such as a river or ocean. In our case, the bottom of the water body is described by the equation z = -h(x), which indicates the depth h as a function of the horizontal position x. Here, the free surface of the water is considered to be at zero (the reference level), meaning everything below this level is negative in height (h). Thus, z decreases with respect to h as we move downward from the water surface.

Imagine standing by the edge of a pool. When the water surface is at the edge, we can think of '0' level being at the top of the water. If you dive down, the depth increases negatively - for instance, at 1 meter below the surface would be z = -1. This analogy helps visualize how we determine depth and reference for studying underwater mechanics.

Since from the boundary condition we see the bottom is fixed for you dot n is going to be 0, which we have seen in the previous lecture. So, we can ride the surface equation as so the equation was z = - h of x. So, we can write 0 + h of x 1 0 and we call it as f function of x and z.

In fluid mechanics, when we refer to a 'fixed bottom condition', we mean that the velocity of the fluid at the bottom boundary is zero. Mathematically, we express this as the dot product of the velocity vector (u) and the outward normal vector (n) being zero (u · n = 0). This suggests that there is no vertical flow at the bottom of the body of water. Essentially, this condition helps in setting the foundation for other calculations related to flow direction and pressure.

Think of the sand at the bottom of a beach. As waves approach, while water moves towards the shore, the sand remains motionless under the water. Just like the sand stays fixed in place, the fluid velocity at the bottom boundary of a water body is also treated as 'fixed' or 'stationary', leading to calculations based on that stable point.

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

When we consider a horizontal bottom, the gradient or change in height with respect to the horizontal distance (dh/dx) is zero. In simpler terms, the bottom does not slope - it remains flat, which implies that the vertical velocity (w) is also zero. Thus, no vertical flow occurs at that boundary. This condition simplifies calculations, as we only need to focus on horizontal flows.

Visualize a large body of water, like a lake, where the bottom surface is flat, akin to a swimming pool. In this scenario, if standing on the bottom, you’d observe that water moves horizontally around you but does not rise or fall vertically - similar to how zero width changes simplifies our computational tasks in fluid dynamics.

Now, 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.

On a sloped bottom, the relationship between the vertical velocity (w) and the horizontal velocity (u) can be expressed with a simple equation: w = -u (dh/dx). This indicates that the vertical flow is inversely proportional to the change in height (dh/dx). In simpler terms, if water flows faster horizontally, it generates corresponding vertical movements, according to the slope's steepness.

Imagine a water slide at a park. The steeper the slide (slope), the faster you move downwards (vertical flow) once you start sliding! This resembles how water behaves over a sloped bottom, where horizontal speed affects vertical motion based on the incline.

Another term is kinematic free surface see, first we studied the boundary condition, we studied kinetic boundary condition in detail. So, the first sub part of that was we saw the bottom boundary condition where the boundary was fixed, where we utilized u dot n = 0. Now, there is something called dynamic free surface boundary condition, free surface means, free surface of water is written as.

The dynamic free surface boundary condition refers to how the free surface of water behaves under various influences, not just being static. While static conditions define a fixed base, a dynamic analysis includes variations over time and space. The motion at the free surface, described as displacement, requires precise calculations to determine how pressure and velocity interact, represented as a function of displacement over time.

Think of ocean waves crashing on the shore. The water surface is not fixed but constantly moves and changes, creating waves. This transition and fluctuation at the surface represent dynamic conditions, similar to how we analyze fluid behaviors in engineering scenarios.

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