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[1] and is broadly classified into two types, bounded and unbounded.[2] 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
\delta(x)
The boundary layer thickness,
\delta
ue
y99
\delta99
u(x,y99)=0.99ue(x) ,
\delta99
\delta99(x) ≈ 5.0\sqrt{{\nux}\overu0}=5.0{x\over\sqrt{Rex}} ,
ue ≈ u0
Rex
u0
ue
x
\nu
\delta
\delta(x) ≈ 0.37{x\over
1/5 | |
{Re | |
x} |
This turbulent boundary layer thickness formula assumes 1) the flow is turbulent right from the start of the boundary layer and 2) the turbulent boundary layer behaves in a geometrically similar manner[5] (i.e. the velocity profiles are geometrically similar along with the flow in the x-direction, differing only by scaling parameters in
y
u(x,y)
The displacement thickness,
\delta1
\delta*
ue
The displacement thickness essentially modifies the shape of a body immersed in a fluid to allow, in principle, an inviscid solution if the displacement thicknesses were known a priori.
The definition of the displacement thickness for compressible flow, based on mass flow rate, is
{\delta1(x)}=
H/2 | |
\int | |
0 |
{\left(1-{\rho(x,y)u(x,y)\over\rhoeue(x)}\right)dy} ,
where
\rho(x,y)
{\delta1(x)}=
H/2 | |
\int | |
0 |
{\left(1-{u(x,y)\overue(x)}\right)dy} .
For turbulent boundary layer calculations, the time-averaged density and velocity are used.
For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the displacement thickness is[7]
\delta1(x) ≈ 1.72\sqrt{{\nux}\overu0} ,
ue ≈ u0
\delta1 ≈ \delta/3
The momentum thickness,
\theta
\delta2
ue
The momentum thickness definition for compressible flow based on the mass flow rate is[10] [11] [12]
\delta2(x)=
H/2 | |
\int | |
0 |
{{\rho(x,y)u(x,y)\over\rhoeue(x)}{\left(1-{u(x,y)\overue(x)}\right)}}dy .
For incompressible flow, the density is constant so that the definition based on volumetric flow rate becomes
\delta2(x)=
H/2 | |
\int | |
0 |
{{u(x,y)\overue(x)}{\left(1-{u(x,y)\overue(x)}\right)}}dy ,
where
\rho
ue
For turbulent boundary layer calculations, the time averaged density and velocity are used.
For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the momentum thickness is[13]
\delta2(x) ≈ 0.664\sqrt{{\nux}\overu0} ,
ue ≈ u0
The momentum thickness is not directly related to the boundary layer thickness but is given approximately as
\delta2 ≈ \delta/6
A related parameter called the Energy Thickness[15] is sometimes mentioned in reference to turbulent energy distribution but is rarely used.
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
H12(x)=
\delta1(x) | |
\delta2(x) |
,
where
H12
\delta1
\delta2
Conventionally,
H12
H12
H12 ≈
H12
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 boundary layer moment method was developed from the observation that the plot of the second derivative of the Blasius boundary layer for laminar flow over a plate looks very much like a Gaussian distribution curve. The implication of the second derivative Gaussian-like shape is that the velocity profile shape for laminar flow is closely approximated as a twice integrated Gaussian function.[20]
The moment method is based on simple integrals of the velocity profile that use the entire profile, not just a few tail region data points as does
\delta99
It is straightforward to cast the properly scaled velocity profile and its first two derivatives into suitable integral kernels.
The central moments based on the scaled velocity profiles are defined as
{\zetan(x)}=
H/2 | |
\int | |
0 |
{(y-m(x))n{1\over\delta1(x)}\left(1-{u(x,y)\overue(x)}\right)dy} ,
where
\delta1(x)
m(x)
m(x)=
H/2 | |
\int | |
0 |
{y{1\over\delta1(x)}\left(1-{u(x,y)\overue(x)}\right)dy} .
There are some advantages to also include descriptions of moments of the boundary layer profile derivatives with respect to the height above the wall. Consider the first derivative velocity profile central moments given by
{\kappan(x)}=
H/2 | |
\int | |
0 |
{
n | |
(y-{\delta | |
1(x)}) |
{d\{u(x,y)/ue(x)\}\overdy}dy} ,
where the first derivative mean location is the displacement thickness
\delta1(x)
Finally the second derivative velocity profile central moments are given by
{λn(x)}=
H/2 | |
\int | |
0 |
{
n | |
(y-{\mu | |
1(x)}) |
{d2\{-\mu1(x)u(x,y)/ue(x)\}\overdy2}dy} ,
where the second derivative mean location,
\mu1(x)
{\mu1(x)}={ue(x)\over\left.{
{du(x,y) | |
\upsilon
\tauw(x)
\mu1
The above equations work for both laminar and turbulent boundary layers as long as the time-averaged velocity is used for the turbulent case.
With the moments and the mean locations defined, the boundary layer thickness and shape can be described in terms of the boundary layer widths (variance), skewnesses, and excesses (excess kurtosis). Experimentally, it is found that the thickness defined as
\deltam=m+3\sigmam
\sigmam=\zeta
1/2 | |
2 |
\delta99
Taking a cue from the boundary layer momentum balance equations, the second derivative boundary layer moments,
{λn}
{λn}
{\zetan}
{\kappan}
Calculation of the 2nd derivative moments can be problematic since under certain conditions the second derivatives can become positive in the very near-wall region (in general, it is negative). This appears to be the case for interior flow with an adverse pressure gradient (APG). Integrand values do not change sign in standard probability framework so the application of the moment methodology to the second derivative case will result in biased moment measures. A simple fix[23] is to exclude the problematic values and define a new set of moments for a truncated second derivative profile starting at the second derivative minimum. If the width,
{\sigmav}
For derivative moments whose integrands do not change sign, the moments can be calculated without the need to take derivatives by using integration by parts to reduce the moments to simply integrals based on the displacement thickness kernel given by
{\alphan}(x)=
H/2 | |
\int | |
0 |
{yn\left(1-{u(x,y)\overue(x)}\right)dy} .
\sigmav
\sigmav=
2+2\mu | |
\sqrt{{-\mu | |
1 |
\alpha0}
\gamma1
\gamma1(x)=\kappa3/\kappa
3/2 | |
2 |
=
3 | |
(2\delta | |
1 |
-6\delta1\alpha1+3\alpha2)/(2\alpha1-
2 | |
\delta | |
1 |
)3/2 .
Numerical errors encountered in calculating the moments, especially the higher-order moments, are a serious concern. Small experimental or numerical errors can cause the nominally free stream portion of the integrands to blow up. There are certain numerical calculation recommendations[25] that can be followed to mitigate these errors.
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.[26] The differences between the bounded and unbounded boundary layer was explored in a series of Air Force Reports.[27] [28] [29]
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
\zetan(x)
{λn}
{\zetan}
{\kappan}
The modified
{\zetan}
{\kappan}
{\deltamax
ue
u0
\deltamax
\deltam=\deltamax+3\sigmai
\sigmai
1/2 | |
{\zeta | |
2 |
The modified
{λn}
\deltamax
An example of the modified moments are shown for unbounded boundary layer flow along a wing section in Figure 3.[32] This figure was generated from a 2-D simulation[33] for laminar airflow over a NACA_0012 wing section. Included in this figure are the modified 3-sigma
\deltam
\deltav
\delta99
\deltam/\delta99
\deltav/\delta99
umax
u0
\deltam
\deltav
\delta99
\delta99
The location of the velocity peak, denoted as
\deltamax
\deltamax ≈
4.3 | |
\delta | |
v |
=\mu1
4.3\sigmav
\deltamax
{\sigmai}
\deltamax
A significant implication of the peaking behavior is that the 99% thickness,
\delta99
\delta99
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.