Boundary layer thickness

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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 refers to the thin transition layer between the wall and the bulk fluid flow. The boundary layer concept was originally developed by Ludwig Prandtl and is broadly classified into two types, bounded and unbounded. The differentiating property between bounded and unbounded boundary layers is whether the boundary layer is being substantially influenced by more than one wall. Each of the main types has a laminar, transitional, and turbulent sub type. The two types of boundary layers use similar methods to describe the thickness and shape of the transition region with a couple of exceptions detailed in the Unbounded Boundary Layer Section. The characterizations detailed below consider steady flow but is easily extended to unsteady flow.

Bounded boundary layers is a name used to designate fluid flow along an interior wall such that the other interior walls induce a pressure effect on the fluid flow along the wall under consideration. The defining characteristic of this type of boundary layer is that the velocity profile normal to the wall often smoothly asymptotes to a constant velocity value denoted as ue(x). 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). Examples of this type of boundary layer flow occur for fluid flow through most pipes, channels, and wind tunnels. The 2 D channel depicted in Figure 1 is stationary with fluid flowing along the interior wall with time averaged velocity u(x,y) where x is the flow direction and y is the normal to the wall. The H/2 dashed line is added to acknowledge that this is an interior pipe or channel flow situation and that there is a top wall located above the pictured lower wall. Figure 1 depicts flow behavior for H values that are larger than the maximum boundary layer thickness but less than thickness at which the flow starts to behave as an exterior flow. If the wall to wall distance, H, is less than the viscous boundary layer thickness then the velocity profile, defined as u(x,y) at x for all y, takes on a parabolic profile in the y direction and the boundary layer thickness is just H/2. At the solid walls of the plate the fluid has zero velocity (no slip boundary condition), but as you move away from the wall, the velocity of the flow increases without peaking, and then approaches a constant mean velocity ue(x). This asymptotic velocity may or may not change along the wall depending on the wall geometry. The point where the velocity profile essentially reaches the asymptotic velocity is the boundary layer thickness. The boundary layer thickness is depicted as the curved dashed line originating at the channel entrance in Figure 1. It is impossible to define an exact location at which the velocity profile reaches the asymptotic velocity. As a result, a number of boundary layer thickness parameters, generally denoted as , are used to describe characteristic thickness scales in the boundary layer region. Also of interest is the velocity profile shape which is useful in differentiating laminar from turbulent boundary layer flows. The profile shape refers to the y behavior of the velocity profile as it transitions to ue(x).

A shape factor is used in boundary layer flow to help to differentiate laminar and turbulent flow. It also shows up in various approximate treatments of the boundary layer including the Thwaites method for laminar flows. The formal definition is given by

where is the shape factor, is the displacement thickness and is the momentum thickness. Conventionally, = 2.59 (Blasius boundary layer) is typical of laminar flows, while = 1.3 1.4 is typical of turbulent flows near the laminar turbulent transition. For turbulent flows near separation, 2.7. The dividing line defining laminar transitional and transitional turbulent values is dependent on a number of factors so it is not always a definitive parameter for differentiating laminar, transitional, or turbulent boundary layers.

Unbounded boundary layers, as the name implies, are typically exterior boundary layer flows along walls (and some very large gap interior flows in channels and pipes). Although not widely appreciated, the defining characteristic of this type of flow is that the velocity profile goes through a peak near the viscous boundary layer edge and then slowly asymptotes to the free stream velocity u0. An example of this type of boundary layer flow is near wall air flow over a wing in flight. The unbounded boundary layer concept is depicted for steady laminar flow along a flat plate in Figure 2. The lower dashed curve represents the location of the maximum velocity umax(x) and the upper dashed curve represents the location where u(x,y) essentially becomes u0, i. e. the boundary layer thickness location. For the very thin flat plate case, the peak is small resulting in the flat plate exterior boundary layer closely resembling the interior flow flat channel case. 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. The differences between the bounded and unbounded boundary layer was explored in a series of Air Force Reports. The unbounded boundary layer peak means that some of the velocity profile thickness and shape parameters that are used for interior bounded boundary layer flows need to be revised for this case. Among other differences, the laminar unbounded boundary layer case includes viscous and inertial dominated regions similar to turbulent boundary layer flows.

For exterior unbounded boundary layer flows, it is necessary to modify the moment equations to achieve the desired goal of estimating the various boundary layer thickness locations. The peaking behavior of the velocity profile means the area normalization of the moments becomes problematic. To avoid this problem, it has been suggested that the unbounded boundary layer be divided into viscous and inertial regions and that the boundary layer thickness can then be calculated using separate moment integrals specific to that region. That is, the inner viscous region of laminar and turbulent unbounded boundary layer regions can be tracked using modified moments whereas the inertial boundary layer thickness can be tracked using modified and moments. The slow rate at which the peak asymptotes to the free stream velocity means that the calculated boundary layer thickness values are typically much larger than the bounded boundary layer case. The modified and 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 , 2) changing the upper integral limit to h where h is located deep in the free stream, and 3) changing the velocity scale from to The displacement thickness in the modified moments must be calculated using the same integral limits as the modified moment integrals. By taking as the mean location, the modified 3 sigma boundary layer thickness becomes where is the modified width. The modified second derivative moments can be calculated using the same integrals as defined above but with replacing H/2 for the upper integral limit. To avoid numerical errors, certain calculation recommendations should be followed. The same concerns for the second derivative moments in regards to APG bounded boundary layers for the bounded case above also apply to the modified moments for the unbounded case. An example of the modified moments are shown for unbounded boundary layer flow along a wing section in Figure 3. This figure was generated from a 2 D simulation for laminar airflow over a NACA 0012 wing section. Included in this figure are the modified 3 sigma , the modified 3 sigma , and the locations. The modified ratio value is 311, the modified ratio value is ~2, and the value is 9% higher than the value. The large difference between the and compared to the value demonstrates the inadequacy of the boundary layer thickness. Furthermore, the large velocity peak demonstrates the problem with treating interior bounded boundary layers as equivalent to exterior unbounded boundary layers.

The location of the velocity peak, denoted as is an obvious demarcation location for the unbounded boundary layer. The main appeal of this choice is that this location is approximately the dividing location between the viscous and inertial regions. For the laminar flow simulation along a wing, umax located at δmax is found to approximate the viscous boundary layer thickness given as + indicating the velocity peaks just above the viscous boundary layer thickness δv. For the inertial regions of both laminar and turbulent flows, is a convenient lower boundary for the moment integrals. If the width, , is calculated using 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.

A significant implication of the peaking behavior is that the 99% thickness, , is NOT recommended as a thickness parameter for the exterior flow, unbounded boundary layer since it no longer corresponds to a boundary layer location of consequence. It is only useful for unbounded laminar flow along a very thin flat plate at zero incidence angle to the flow direction since the peak for this case will be very small and the velocity profile will be closely approximated as the bounded boundary layer case. For thick plates walls, non zero incidence angles, or flow around most solid surfaces, the excess flow due to form drag results in a near wall peak in the velocity profile making not useful.

The displacement thickness, momentum thickness, and shape factor can, in principle, all be calculated using the same approach described above for the bounded boundary layer case. However, the peaked nature of the unbounded boundary layer means the inertial section of the displacement thickness and momentum thickness will tend to cancel the near wall portion. Hence, the displacement thickness and momentum thickness will behave differently for the bounded and unbounded cases. One option to make the unbounded displacement thickness and momentum thickness approximately behave as the bounded case is to use umax as the scaling parameter and δmax as the upper integral limit.