2.5. Equation for Displacement Thickness

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 ( delta dash,  δ*) is a concept in fluid dynamics that quantifies how much the flow is altered near a wall due to friction. It can be imagined as the distance at which a hypothetical streamline, which would not experience any viscous effects, would be shifted away from the wall due to the presence of a boundary layer.

Think of a rubber band being stretched out by pulling on both ends. If the rubber band represents the flow of fluid, pulling it outwards at one end (the wall) will displace the entire structure slightly away from its original position—similar to how the fluid is altered and displaced around the plate.

Now, we consider the flow over a smooth flat plate... The strip will have the height dy and the width is b. Therefore, the area is simply bdy.

When examining the flow over a flat plate, we consider a small vertical strip (elemental strip) of thickness dy at a distance y from the plate. The area of this strip is calculated as width b multiplied by height dy. This area is crucial for calculating the mass flux, which quantifies how much fluid mass passes through this area per unit time.

Imagine water flowing in a narrow stream. If we slice through the stream vertically at a certain point, we can analyze how much water passes through this 'slice' by multiplying the width of the stream with the height of the water at that specific point, similar to our fluid analysis.

Therefore, the reduction in the mass flux through the elemental strip... ρ U - u into bdy.

In this step, we compare the mass flux through our elemental strip with a hypothetical scenario where the fluid exhibited uniform velocity throughout the layer. The difference in these mass fluxes gives us insights into how much mass is 'lost' due to the boundary layer effects, which can be quantified by the expression ρ(U - u). This difference directly correlates with the impact that the viscous layer has on the overall fluid flow.

Consider a highway where cars (fluid molecules) are moving at different speeds due to congestion (viscous effects). The difference between the maximum speed allowed on the highway and the actual speed of the cars in traffic represents a reduction in the mass flux of cars reaching a certain point, akin to our mass flux analysis.

Thus, for an incompressible fluid, we obtain, delta dash is equal to... for the displacement thickness.

To derive the equation for displacement thickness, we integrate the mass flux reductions over the thickness of the boundary layer. This process gives us a concrete mathematical representation of how fluid displacement due to viscous effects can be quantified. The final equation demonstrates the relationship between the flow characteristics and the displacement thickness, allowing engineers to calculate it under various conditions.

Imagine pouring a thick syrup (viscous fluid) over a table (boundary layer). As you pour, the syrup spreads out, creating a thicker layer on the table compared to thinner layers further away. By measuring the depth of the syrup at various points, we can derive an equation describing how the syrup 'displaces' normal air flow above it—a process akin to calculating displacement thickness in fluid dynamics.

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