4.1. Flat Plate Dynamics

When a real fluid flows past a solid, the fluid particles stick to the solid surface. The velocity of the fluid particles close to the solid boundary is equal to the velocity of the boundary, called the no slip boundary condition. For a stationary body, the fluid velocity at the boundary is zero.

The no slip boundary condition states that when a fluid flows near a solid surface, the fluid's velocity at the point of contact (the boundary) will be equal to the surface's velocity. If the surface is stationary, the fluid's velocity at that point is zero. This means that any fluid particle in direct contact with a still surface essentially 'sticks' to it, leading to a change in velocity just above the surface.

Imagine a ball rolling on a smooth surface. The very top layer of the ball in contact with the ground comes to a complete stop due to friction — that’s similar to fluid particles sticking to a solid surface.

This velocity variation occurs in a very thin region of flow near the solid surface, called the boundary layer. The fluid velocity increases from zero at the solid surface to the free-stream velocity in the direction normal to the boundary.

The boundary layer is a region that forms along a solid surface when a fluid flows over it. Within this layer, the velocity changes from being zero at the boundary (due to the no slip condition) to nearing the free stream velocity as you move away from the surface. The thickness of this layer is small compared to the overall flow depth, but it is crucial for understanding how viscosity affects flow behavior.

Think of the way your hand feels when moving it through water quickly. Right at your hand, where the water touches it, the motion is almost still because of the viscosity, while a bit further away, the water flows freely and quickly.

Prandtl divided the flow into two regions: the boundary layer, where viscous forces cannot be ignored, and the outer flow region, where the velocity is constant at free stream velocity. The fluid exerts shear stress on the wall, which is proportional to the velocity gradient.

According to Prandtl, the behavior of fluid flow near a solid surface can be classified into two areas. In the boundary layer, the effects of viscosity create shear stresses, while in the outer flow region, there are negligible effects from viscosity, and the flow is primarily determined by inertial forces, resulting in a constant velocity equal to the free stream. The shear stress arises due to the velocity gradient in the boundary layer.

Imagine a thick blanket of air surrounding you while you ride a bike. Close to your skin (like the boundary layer), the air is affected by your body and doesn’t move fast — that’s where the shear stress occurs. Further away from your body, the air moves with the same speed regardless of your bike’s speed.

The boundary layer grows as fluid flows downstream from the leading edge of the flat plate, with a thickness that increases with distance along the plate.

As fluid flows past the leading edge of a flat plate, the boundary layer begins to form and grow thicker downstream. The distance along the plate is referred to as 'x,' and as you move further along the flat plate, the boundary layer's thickness continues to increase due to the establishment of velocity gradients and shear stresses in the flow.

Consider how a snowball grows as you roll it on the snow; it collects more snow and gets bigger the further you roll it. Similarly, the boundary layer thickens as the fluid flows further over the flat plate.

Near the leading edge, the flow in the boundary layer is laminar, and the thickness is stable up to a Reynolds number of 5 x 10^5. Beyond this number, the laminar flow becomes unstable, leading to turbulence.

In the initial region along the flat plate, the fluid flow in the boundary layer is characterized as laminar, meaning the flow is smooth and orderly. However, as velocity increases with distance (represented as Reynolds number), when the Reynolds number exceeds 5 x 10^5, the flow becomes unstable and transitions into turbulent flow characterized by chaotic fluctuations and mixing.

Think of a quiet stream where the water flows smoothly around rocks (laminar flow). As more water rushes in and the current speeds up, things get choppy and never settle down, just like the flow when it gets turbulent.

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