Using numerical results obtained by Blasius (Table 9.1),...

Using numerical results obtained by Blasius (Table 9.1), evaluate the distribution of shear stress in a laminar boundary layer on a flat plate. Plot $\tau / \tau_{w}$ versus $y / \delta .$ Compare with results derived from the approximate sinusoidal velocity profile given in Problem 9.10.

Key Concepts

Laminar Boundary Layer

A laminar boundary layer is the thin region adjacent to a solid surface, such as a flat plate, where viscous forces dominate and the fluid flow exhibits a smooth, orderly behavior. In this region, the velocity increases gradually from zero at the wall (due to the no-slip condition) to the free stream value, resulting in specific velocity gradients that are central to the analysis of skin friction and momentum diffusion.

Blasius Solution

The Blasius solution is a similarity solution of the boundary layer equations for steady, incompressible, laminar flow over a flat plate. This numerical solution provides detailed information about the velocity profiles within the boundary layer and enables the calculation of important quantities, such as the skin friction (or wall shear stress) and its distribution, by reducing the partial differential equations to an ordinary differential equation using appropriate scaling and similarity variables.

Wall Shear Stress Distribution

Wall shear stress is a measure of the tangential force per unit area exerted by the fluid as a result of viscosity and is directly related to the velocity gradient at the wall. In boundary layer theory, evaluating the distribution of shear stress across the boundary layer (often normalized with the wall value) helps in understanding the frictional effects and drag characteristics over the surface. It is a key parameter in characterizing the fluid-structure interactions in viscous flows.

Non-Dimensionalization in Fluid Mechanics

Non-dimensionalization involves scaling the governing equations and resultant profiles by characteristic quantities such as the boundary layer thickness and wall shear stress. This process leads to dimensionless parameters (for example, y/? and ?/?w) that allow the analysis to be generalized and compared across different flow conditions. It simplifies the interpretation of results and facilitates the comparison between numerical solutions like Blasius' and approximate analytical forms such as the sinusoidal velocity profile.

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