10.2.3. Navier-Stokes Equations Recap

So basically let us start with Navier-Stokes equations which is long back about 200 years back. So the equations what we derive in last two classes we will go more detail about that.

The Navier-Stokes equations are fundamental equations in fluid mechanics that describe the motion of fluid substances. They were developed over 200 years ago and have been foundational in understanding how fluids behave in various situations. In this section, we will build upon what was previously covered regarding these equations.

Think of the Navier-Stokes equations as the 'rules of the road' for fluids. Just as traffic rules dictate how cars move, these equations dictate how fluid particles move and interact with each other and their surroundings.

Before going that let me I just write down the basics equations is the Euler equations okay. ... But newer to structures as the vorticity is there, the turbulence behaviors are there as well as it has the flow separations that is what is not valid for this external flow.

The Euler equations are derived from the Navier-Stokes equations under the assumption of inviscid flow (no viscous forces). These equations apply primarily to incompressible, non-viscous flows. In practical scenarios, like wind passing tall buildings, the flow can be approximated with Euler equations, but near the structures, where turbulence and vortices occur, these equations do not hold well.

Imagine a river flowing past a rock. The water flows smoothly without interaction (similar to Euler's assumptions) until it encounters the rock, creating turbulence and whirlpools. This reflects how Euler equations are valid where the flow is smooth and laminar but break down in turbulent areas.

The basic idea comes that can you write these equations with a single scalar value okay that is the simple idea comes is that instead of looking at 3 scalar velocity component why we cannot write it with a simple a scalar functions that is what is the velocity potential function.

The concept of velocity potential functions simplifies the analysis of fluid flow by representing the flow velocity as the gradient of a scalar function, known as the velocity potential. Instead of dealing with three separate components of velocity, which can complicate calculations, a single scalar function can provide the same information in a more manageable way. This approach is particularly useful in irrotational flow conditions.

Consider measuring temperature across a field. Instead of checking the temperature at every point (like measuring each velocity component), you could describe the entire field with a single temperature map. The velocity potential function acts similarly, capturing the essence of the flow in a simplified manner.

Thus if I define it the velocity is a gradient of phi, phi is a velocity potential function... That means if I again coming back to the same flow first a tall building if I draw the two streamlines.

The irrotational flow condition is key to applying velocity potential functions. It states that in regions where the fluid behaves without rotation, the velocity can be represented as a gradient of a potential function. When taking flow measurements around structures, the flow may become rotational; thus, the conditions of irrotational flow are essential for accurate calculations using potential functions.

Imagine a calm lake surface reflecting the sky. This reflects an irrotational condition—the water flows smoothly without any disturbances. However, if a stone is thrown into the lake, waves and vortices form, creating an 'irrotational' disruption to that calm state.

We need to draw the streamlines and the potential lines to show it that because just you interpreted the gradient of the velocity potential functions indicates as the velocity field that is very interesting part of here.

Streamlines and velocity potential lines visualize fluid flow, with streamlines representing the path of fluid motion and velocity potential lines illustrating the potential function. Both types of lines are crucial for understanding flow patterns, and their orthogonality signifies that they intersect at right angles, which is essential in two-dimensional irrotational flow.

Think of a road map with streets (streamlines) and altitude lines (potential lines). The streets show where cars can go while the altitude lines indicate heights. In this analogy, cars follow streets smoothly, just as fluid particles follow streamlines, while altitude lines help visualize energy transitions in the flow.

Now if you look at very interesting part which I can relate it with very basic equations what you know it is the Newton's first law of motions is you know it if you look at that Newton's laws first law of motions okay you know with the three laws okay.

This section correlates the concept of rotationality in fluid dynamics with Newton's first law of motion. It explains that just as a body remains at rest or moves uniformly in a straight line unless acted upon by an external force, an irrotational flow remains unchanged without external influences. When disturbances occur, like obstacles in the flow path, they create rotation within the fluid.

Imagine riding a bicycle on a straight flat road. If you keep pedaling evenly without any obstacles, you move smoothly (irrotationality). If you suddenly hit a curb (external force), the bike's direction and flow (just like fluid behavior) change, causing rotational behavior.

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