Assume a laminar boundary layer for a fluid can be...

Assume a laminar boundary layer for a fluid can be approximated by $u / U=\sin (\pi y / 2 \delta) .$ Determine the displacement thickness $\delta^{\prime \prime}$ for the boundary layer as a function of $x$ and $\operatorname{Re}_{x}$.

Key Concepts

Laminar Boundary Layer

A laminar boundary layer is a thin region adjacent to a solid surface where viscous forces dominate and the flow is smooth and orderly. In this region, fluid particles follow parallel paths without the random mixing seen in turbulent flows. This concept is crucial when analyzing velocity profiles and understanding mass and momentum transfer near surfaces.

Displacement Thickness

Displacement thickness is a measure of the effective shift of the outer flow due to the presence of the boundary layer. It quantifies the deficit in mass flow caused by the slower velocities within the boundary layer. Mathematically, it is defined by an integral that measures the reduction of flow compared to an ideal, inviscid flow, and it is important for predicting the influence of the boundary layer on the overall fluid flow past a surface.

Velocity Profile Approximation

The velocity profile approximation in a boundary layer problem represents how the velocity varies from the wall to the free stream. Using a prescribed function, such as a sinusoidal profile, facilitates analytical integration to obtain key displacement properties. This approximation is critical as it allows the derivation of expressions for quantities like the displacement thickness.

Reynolds Number

The Reynolds number is a dimensionless parameter that compares inertial forces to viscous forces in a fluid flow. In boundary layer theory, the local Reynolds number, defined as Re_x = Ux/?, is used to characterize the flow and to scale the boundary layer thickness with respect to the distance from the leading edge. This parameter is central to predicting and understanding the behavior of laminar flows over surfaces.

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