13.6.2. Apparent Wall Concept

Now if you look at the understanding of boundary layers, talk about how the variations of wall shear stress or screen factors. Now if you look it then another concept what we look it called a displacement thickness. The basically as we know it the boundary layer thickness developments happens it okay. The regions which is closer to that boundary layer formations are happening it.

In fluid mechanics, as flow passes over a surface, a boundary layer forms where the flow velocity changes from zero at the surface (due to the no-slip condition) to the free stream velocity. The displacement thickness is a measure of how much the flow is affected by the boundary layer, effectively displacing the external flow. This concept is important as it helps us understand how the formation of boundary layers alters the flow characteristics around objects.

Imagine a river flowing around a rock. Near the surface of the rock, the water moves slowly or almost not at all (that's the boundary layer), while further away, the water flows swiftly (that's the free stream). The displacement thickness quantifies how much the water's line of flow is shifted outward due to the slower velocity near the rock.

So the basic idea is comes it as the boundary layer thickness happens it and if I take a just a free out streamlines okay just of a free streamlines okay which I can consider this streamlines without boundary layer formations okay that is the without the boundary layer formations the streamline could go on like this S1 conditions.

Displacement thickness is defined as the distance the streamline would be pushed outward due to the presence of the boundary layer. When a boundary layer forms, the actual streamline paths deviate from where they would be if the boundary layer didn't exist. This displacement can be calculated quantitatively, helping engineers design systems that account for these shifts in flow patterns.

Think of a swimming pool where someone moves their arms while swimming. The disturbance caused by their arms creates ripples that move outward. These ripples would represent the displacement of water flow due to the swimmer's presence, much like how the displacement thickness displaces external flow due to the boundary layer.

So if you look at this is what displacement thickness. So basically this is mass conservation equations concept that we are talking about the mass what will be because of deflecting of these streamlines will be delta star is a displacement thickness into the u that is what will be 0 to y or we can put the infinities okay.

Mathematically, the displacement thickness can be expressed using the integral form over the velocity distribution within the boundary layer. This relationship is derived from the principles of mass conservation, showing how the velocity deficit across the boundary layer results in an effective outward shift of the streamline. It quantifies the impact of boundary layer formation on flow rates.

Consider trying to squeeze a hose full of water. The constriction is like the boundary layer, and while water is still flowing, the streamlines are being deflected outward. The distance that the water is pushed out from its usual path while passing through the constriction can be likened to the displacement thickness.

So this is what helps this displacement thickness is to know it we are creating a if I try to understand it that means we have an actual wall okay that is the reasons we put it this is actual wall. but the displacement thickness is gives is as a apparent walls okay.

The apparent wall concept arises from the need to analyze flows as if they behave in a simplified, ideal manner. The idea is that the displacement thickness behaves like an imaginary wall where the impact of the boundary layer can be considered. This allows fluid dynamicists to work with equations that govern irrotational flow, which are often simpler than those for viscous flows.

Think of a virtual wall made of air located just outside where the actual wall is. While the surface of the wall is still present, the flow is imagined to act as though it is interacting with this virtual wall instead. This is similar to how a strong wind moves around a building; the building pushes the air outward as if there were an invisible barrier.

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