1.4. Lecture – 18

Displacement thickness is the distance by which a streamline, just outside the boundary layer, is displaced away from the wall due to viscous effects on the plate.

Displacement thickness is an essential concept in boundary layer theory. It describes how much a streamline is pushed away from a wall by viscous forces in the fluid. When a fluid flows over a flat plate, it experiences friction due to its viscosity, causing slower velocities near the plate compared to the free stream. Displacement thickness quantifies this effect and is important for understanding how much the flow is altered due to the presence of the plate.

Imagine a large crowd of people trying to walk through a narrow doorway. As they approach the doorway, some people slow down due to congestion. The displacement thickness is analogous to how far the crowd is pushed back from the doorway. The higher the congestion (or viscosity in fluid terms), the more the crowd spreads out before reaching that doorway.

Now, we consider the flow over a smooth flat plate, like this... The area of the strip can be given by, b into dy...

To derive displacement thickness, we consider a flat plate with fluid flowing over it. We look at an elemental strip of thickness 'dy'. The mass flux through this strip is calculated as the product of density and velocity across that area. When we compare the mass flux in this boundary layer with that in an unrestricted flow (or uniform flow), we find the difference is due to the reduction in velocity caused by the viscous effect. This difference, when integrated across the boundary layer, gives us the displacement thickness.

Think of water flowing through a wide pathway that narrows into a vent. As the water approaches the narrow opening, it slows down and spreads out. This 'spreading out' effect can be calculated in a similar way to how we derive the displacement thickness in fluid dynamics.

Momentum thickness theta is the loss of momentum flux in the boundary layer, as compared to that of the potential flow.

Momentum thickness quantifies the loss of momentum in a boundary layer compared to an ideal flow without any boundary effects. It is defined through momentum flux, which is the mass flux multiplied by velocity. As fluid flows over the boundary, some of its momentum is lost due to friction with the surface, causing a deficit that represents our momentum thickness.

Imagine a skateboarder gliding smoothly along a flat sidewalk (ideal flow), but then transitioning onto a rougher surface like gravel (boundary layer effects). The skateboarder would lose speed and momentum due to friction with the gravel. The difference in speed they experience when gliding on gravel versus on the smooth sidewalk is akin to momentum thickness.

Now, there is something called energy thickness, delta double dash. This is defined based on the reduction of the kinetic energy of the fluid flow due to the velocity defect.

Energy thickness is another way of looking at the effects of viscosity in a fluid flow. It focuses on how the kinetic energy of a flow is reduced due to the slower speeds in the boundary layer compared to the free stream. This reduction in energy can greatly affect the energy transfer and performance in various engineering applications.

Consider a car trying to go uphill. The car has to expend more energy to overcome gravitational forces. Similarly, when fluid flows over a surface and encounters resistance (due to viscosity), it loses kinetic energy, which we can measure as energy thickness.

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