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This is considered one of the simplest unsteady problems that has an exact solution for the Navier–Stokes equations. [ 1 ] [ 2 ] In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments , numerical simulations or approximate methods in order to obtain useful information on the flow.
The basis of the Falkner-Skan approach are the Prandtl boundary layer equations. Ludwig Prandtl [2] simplified the equations for fluid flowing along a wall (wedge) by dividing the flow into two areas: one close to the wall dominated by viscosity, and one outside this near-wall boundary layer region where viscosity can be neglected without significant effects on the solution.
The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer, while the Blasius boundary layer refers to the well-known similarity solution near an attached flat plate held in an oncoming unidirectional flow and Falkner–Skan boundary layer, a generalization of Blasius profile.
A schematic diagram of the Blasius flow profile. The streamwise velocity component () / is shown, as a function of the similarity variable .. Using scaling arguments, Ludwig Prandtl [1] argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate).
In this article the term "flow" is used interchangeably with the term "solution" due to historical reasons. The techniques involved to create more complex solutions can be for example by superposition , by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or boundary layer and to be patched ...
The boundary layer thickness, , is the distance normal to the wall to a point where the flow velocity has essentially reached the 'asymptotic' velocity, .Prior to the development of the Moment Method, the lack of an obvious method of defining the boundary layer thickness led much of the flow community in the later half of the 1900s to adopt the location , denoted as and given by
An advantage of the plug flow model is that no part of the solution of the problem can be perpetuated "upstream". This allows one to calculate the exact solution to the differential equation knowing only the initial conditions. No further iteration is required. Each "plug" can be solved independently provided the previous plug's state is known.
Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (for example, the Blasius boundary layer of an infinite plate, but not of a finite-length plate). These include, for example, the Blasius boundary layer or the Sedov–Taylor shell. [1] [2]