3.1. Dynamic Free Surface Boundary Condition

Now, there is something called dynamic free surface boundary condition, free surface means, free surface of water is written as. So, you the surface is free it can distort if you recall from our open channel flow lectures.

The dynamic free surface boundary condition describes how the surface of water behaves when disturbed. Unlike a fixed boundary, a free surface can change or distort due to factors like waves or fluid motion. This means that the boundary at the surface is not constant and can rise or fall based on the fluid dynamics involved.

Think of a water balloon; when you poke it or push on it, the surface of the balloon changes shape. Similarly, the surface of water in a river or ocean changes shape due to waves or currents.

So, if the surface is something like this, you know then the free surface and this is it as a function of x y and t, then we can write F of x y z, t = z – eta x y t.

To describe how the free surface of water fluctuates over time and space, we use a mathematical function, F(x,y,z,t). This function accounts for the vertical position (z) of any point on the surface, minus the water's free surface elevation (eta), which can vary based on location (x, y) and time (t). It essentially measures the distance from the free surface at any point.

Imagine tracking the height of ocean waves along a coastline. You can think of F as a way to measure how high the water rises and falls at each point along the shore throughout the day.

So delta F is going to be delta F delta x i cap + delta F delta y Jacob + delta F delta z k cap, so, if we put F as here z - eta x y t, then we get – del eta del x – del eta del y + k cap.

In order to analyze the movement of the free surface, we calculate the change in F (ΔF) in all three spatial dimensions - x, y, and z. By substituting our earlier equation into this calculation, we can determine how the surface elevation changes relative to those dimensions, leading to rates of change for eta in both horizontal directions (x and y).

Imagine you're at a pool watching how water levels rise or fall at different spots. If someone jumps in or splashes, the waves will change at various places (x and y directions), and you'll have to observe how each point in the pool reacts at the same time (z direction).

So, this is the equation of the free surface this is an important equation. So, this is the dynamic free surface boundary condition.

The derived dynamic free surface boundary condition is an essential equation that relates the water's surface velocity to its change over time. This equation incorporates not just the water motion along the surface but also how that motion is influenced by waves, ultimately describing how a free surface can dynamically change shape.

Think about a soccer player dribbling a ball on a wet grassy field. The way the ball moves across the surface (representing the water surface) can change rapidly, depending on how the player interacts with it (similar to how water reacts to wind or other forces).

Therefore another boundary condition is required for any free surface or interface to prescribe the pressure distribution and the boundary. Since the free surface cannot support the pressure variation, we need to prescribe another boundary condition to tell what the pressure distribution on that particular boundary is and this is called the dynamic boundary condition.

Unlike fixed boundaries, where the pressure changes can be managed and calculated, free surfaces can only support uniform pressure along their length because they can deform. This requirement leads to the formation of additional boundary conditions to properly describe the pressure distribution atop a fluid's surface, known as dynamic boundary conditions.

Consider a trampoline; while jumping, the surface does not hold pressure variations well since it bends and shifts under weight. Instead, everyone on the trampoline experiences a roughly uniform bounce effect, akin to how water's free surface behaves under surface tension.

To derive the dynamic free surface boundary condition we use unsteady Bernoulli’s equation on the free surface.

To accurately characterize the dynamics of the free surface, especially for fluids in motion, we apply Bernoulli's equation adjusted for changing conditions (unsteady), which helps illustrate how fluid pressure, velocity, and elevation interrelate at the free surface. This application forces us to consider both time and space changes, making the analysis richer and complex.

Imagine riding a roller coaster; your experience changes as you go up and down hills. Similarly, applying the unsteady Bernoulli equation helps us understand how the speed and pressure of the water changes as speed varies depending on surface waves.

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