Boundary layer thickness
This page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface.
The defining characteristic of boundary layer flow is that at the solid walls, the fluid's velocity is reduced to zero.
The boundary layer concept was originally developed by Ludwig Prandtl[1] and is broadly classified into two types, bounded and unbounded.
The bounded boundary layer concept is depicted for steady flow entering the lower half of a thin flat plate 2-D channel of height H in Figure 1 (the flow and the plate extends in the positive/negative direction perpendicular to the x-y-plane).
The boundary layer thickness is depicted as the curved dashed line originating at the channel entrance in Figure 1.
Also of interest is the velocity profile shape which is useful in differentiating laminar from turbulent boundary layer flows.
For laminar boundary layer flows along a flat plate channel that behave according to the Blasius solution conditions, the
Neither one of these assumptions is true for the general turbulent boundary layer case so care must be exercised in applying this formula.
, is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity
For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the displacement thickness is[7] where
, is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity
For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the momentum thickness is[13] where
It also shows up in various approximate treatments of the boundary layer including the Thwaites method for laminar flows.
values is dependent on a number of factors so it is not always a definitive parameter for differentiating laminar, transitional, or turbulent boundary layers.
A relatively new method[18][19] for describing the thickness and shape of the boundary layer uses the mathematical moment methodology which is commonly used to characterize statistical probability functions.
The moment method introduces four new parameters that help describe the thickness and shape of the boundary layer.
Applying the moment method to the first and second derivatives of the velocity profile generates additional parameters that, for example, determine the location, shape, and thickness of the viscous forces in a turbulent boundary layer.
[21] It is straightforward to cast the properly scaled velocity profile and its first two derivatives into suitable integral kernels.
is given by There are some advantages to also include descriptions of moments of the boundary layer profile derivatives with respect to the height above the wall.
track the thickness and shape of that portion of the boundary layer where the viscous forces are significant.
, is calculated using the minimum as the mean location, then the viscous boundary layer thickness, defined as the point where the second derivative profile becomes negligible above the wall, can be properly identified with this modified approach.
Small experimental or numerical errors can cause the nominally free stream portion of the integrands to blow up.
The unbounded boundary layer concept is depicted for steady laminar flow along a flat plate in Figure 2.
This has led much of the fluid flow literature to incorrectly treat the bounded and unbounded cases as equivalent.
The problem with this equivalence thinking is that the maximum peak value can easily exceed 10-15% of u0 for flow along a wing in flight.
[26] The differences between the bounded and unbounded boundary layer was explored in a series of Air Force Reports.
moments for the inertial boundary layer region are created by: 1) replacing the lower integral limit by the location of the velocity peak designated by
An example of the modified moments are shown for unbounded boundary layer flow along a wing section in Figure 3.
For the laminar flow simulation along a wing,[35] umax located at δmax is found to approximate the viscous boundary layer thickness given as
as the mean location then the boundary layer thickness, defined as the point where the velocity essentially becomes u0 above the wall, can then be properly identified.
