Boundary Layers

As a fluid travels across a surface or through a vessel such as a pipe, the contact-making surface will cause a frictional drag upon the adjacent fluid. This frictional force upon the fluid causes the fluid to slow near the obstructing plate's surface, creating a velocity profile within the fluid. Understanding how the velocity profile forms is important in understanding what a boundary layer is. Typically, the boundary layer is defined as a distance y from the x-axis where the local velocity of the liquid is 99% of the free stream velocity. The boundary layer in a fluid moving in a laminar fashion will be much different to that of the same fluid moving in a turbulent fashion and their respective velocity profile's will also be much different. Therefore, the Reynold's number of the fluid should be determined before attempting to calculate a velocity in a moving fluid or boundary thickness in order to understand the behaviour of the fluid.

For a laminar-behaving fluid moving across a flat plate, the drag force and other properties can be determined using the following derivation:

For fluid moving in a turbulent fashion, different empirically derived coefficients are used and using different assumptions. Here are the corresponding coefficients for the equations above but for turbulent flows:

The image on the left shows a dramatic laminar velocity profile in a pipe, typically in most practical situations the laminar flow's boundary layer is insignificant. However, it is easiest to present the concept in laminar fluid flows. The velocity can be found at any point along the y axis using the following derivation:

There are multiple other ways of defining the boundary layer, which have their own derivations depending on which properties are being considered in the fluid. For example, we can understand the boundary layer's thickness as a lost mass to the boundary layer. This is an example of such a derivation: